### Angles formed by the hands of a clock - January 1996

This month's project has three parts:

1. At how many different times will the hands of a clock make a right angle? At what times will this occur? Determine your answers to the nearest second.

2. Find a time at which the hands of a clock make a 45-degree angle. Generalize your method to find a time at which the hands of a clock form any given angle.

3. What angle will the hands of a clock form at 3:20? Do not use a protractor. Generalize your method to give the angle formed by the hands of a clock at any given time.

### Submissions

```From: be301@scn.org  (Seattle Community Network)
Kent Cheung
School: Nathan Hale High School, Seattle, WA.

Note: Although this may indeed be true in actuality, for this
problem I have assumed that all three hands on the clock
(the hour hand, the minute hand, and the second hand) will
overlap at precisely 12:00:00.  That is, the tip of all
three hands will point straight up directly at the number
"12" on the clock.

It can be shown that the hour and minute hands must overlap
each other 11 times in the 12 hours during which the hour hand
moves 1 complete revolution around the clock, starting at 12:00:00
and eventually ending back at 12:00:00.  For example, the hour hand
and the minute hand would overlap at approximately 01:05, and
they would do so again at around 02:10, and again at 03:15, and
so on.  Therefore, it can be concluded that the hands will overlap
once every 12/11 hours, that is, every 1.0909... hours.  This
translates approximately to 1 hour, 5 minutes, and 27.27 seconds.
For example, the hour hand and the minute hand will overlap at
precisely 01:05:27, to the nearest second.  They will overlap again
exactly 1 hr 5 min 27 sec later at 02:10:54 and they will overlap
yet again another 1 hr 5 min 27 sec later at 03:16:21, and so forth.

Before proceeding further, a few key concepts and definitions
must be established:

* The clock is divided into 60 "units," each unit representing
the distance traversed by the second hand in 1 second (or the
distance traversed by the minute hand in 1 minute, or 1/5 the
distance traversed by the hour hand in 1 hour).  For example,
the distance between the hour hand and the minute hand at
12:15:00 is 15 units.

* For every unit the second hand moves, the minute hand moves
1/60 unit (one-sixtieth of a unit).  For every 60 units
(1 revolution) the minute hand moves around the clock, the
hour hand moves 5 units.  Therefore, the hour hand moves 5/60,
or 1/12, of a unit for every 1 unit the minute hand moves.
It follows that for every unit the second hand moves, the hour
hand moves 1/720 of a unit (1/720 = 1/60 * 1/12).

* For the sake of easy reference, I will refer to the number
"12" on top of the clock as the "origin."  For example, at
3:00 the hour hand is exactly 3 units from the origin.

By studying the clock, it can be seen that the hour hand and
the minute hand will form a 90-degree angle at approximately 12:15.
At 12:15:00, the minute hand is exactly 15 units from the origin
and the hour hand is exactly 15/12 units from the origin (recall
that for every 1 unit the minute hand moves, the hour hand moves
1/12 unit).  To find the distance between the hands, we find the
difference between their distances from the origin.  In this case,
the hour hand is 15/12 units from the origin and the minute hand
is 15 units from the origin, so their difference is the absolute
value of (15/12 - 15), or 13.75. (See Table 1).

Table 1.

Distance from the origin

hr.hand  min.hand  sec.hand   Distance between min. & hr. hands
---------------------------------------------------------------
15/12     15         0       ABS(15/12 - 15)  = 13.75

16/12     16         0       ABS(16/12 - 16)  = 14.67

17/12     17         0       ABS(17/12 - 17)  = 15.58

In order for the hands to be at 90 degrees, they must be
exactly 15 units apart.  According to Table 1, the hands are less
than 15 units apart at 12:16:00.   However, at 12:17:00 the hands
are more than 15 units apart.  Therefore, the hands must be "exactly"
15 units apart sometime between 12:16:00 and 12:17:00.

Now, the number of seconds will have to be taken into consideration.
We know that the hands will be at 90 degrees at 12:16:n, where
the n stands for an unknown number of seconds.  As an example,
let n be 5, then the time is 12:16:05. (See Table 2).

Table 2.
Distance from the origin

Time           hr.hand          min.hand   sec.hand
----------------------------------------------------
12:06:00      16/12             16             0

12:06:05      16/12 + 5/720     16 + 5/60      5

At 12:06:05, the minute hand has moved 16 units from the origin,
so the hour hand is 16/12 units from the origin.  But in addition
to this, the second hand has moved 1 unit from the origin, so the
hour hand has moved an additional 1/720 of a unit, a total of
16/12 + 1/720 units from the origin. (See Table 2).  Because the
second hand has moved 1 unit, the minute hand has moved an
additional 1/60 unit, a total of 16 + 1/60 units from the origin.

Now that we know how far the hour and minute hands are from the
origin, the the distance between them can be calculated:

Table 3.

Time        Distance between min. & hr. hands
--------------------------------------------------------------
12:06:00     ABS(16/12 - 16)                          = 14.67

12:06:05     ABS[ (16/12 + 5/720)  -  (16 + 5/60) ]   = 14.74

Instead of 5, let "n" be the number of seconds.  Since the
distance between the hands must be exactly 15 units when they are
at a 90-degree angle, the exactly value of n can be found by
solving the following equation for n:

ABS[ (16/12 + n/720)  -  (16 + n/60) ]  = 15

Solving for n yields 21.8181...

Therefore, the hour and minute hands will be exactly 15 units
apart at 12:16:22, to the nearest second (thus forming an angle of
90 degrees with respect to each other).

12:16:22 is precisely 16 min and 22 sec from 12:00:00.  This is
the same as one-fourth of 1 hr 5 min and 27 seconds, which was
established before as the period of time between overlaps of the
hour and minute hands.

When the hands overlap, it means that they are at an angle of
360 degrees with respect to each other (it may also be
interpreted that they are at an angle of 0 degrees).  If a table
is generated listing the various angles and the associated times,
a pattern may become apparent:

Table 4.

Angle (deg.)   Time      Number of hours from 12:00:00
-------------------------------------------------------
360           01:05:27   1.09 hrs
180 = 360/2   12:32:44   0.54 hrs = 1.09/2 hrs
90 = 360/4   12:16:22   0.27 hrs = 1.09/4 hrs
45 = 360/8   12:08:11   0.14 hrs = 1.09/8 hrs

From the table, it would appear that the number of hours from
12:00:00 is equal to:

1.09/(360/a),         where a is the angle expressed in deg.

Recall that the value 1.09 was actually derived from the fraction
12/11, thus by replacing 1.09 in the equation above with 12/11
we obtain:

(12/11)/(360/a)   Hence: a/330,  where a is expressed in deg.

---------------------------------------------------------------

It can be shown using procedures described previously that
there are 22 different times when the hour and minute hands are
at 90 degrees to each other:

12:16:22  02:27:16  04:05:27  06:16:22  08:27:16  10:05:27
12:49:05            04:38:11  06:49:05            10:38:11

01:21:49  03:00:00  05:10:55  07:21:49  09:00:00  11:10:55
01:54:33  03:32:44  05:43:38  07:54:33  09:32:44  11:43:38

With the exception of 180 degrees, all other angles less than
180 degrees can be formed by the hands of the clock at most 22
different times.  (The hands will form 180 degrees only 11 times.
All angles greater than 180 degrees are identical to some other
angle which are less than 180 degrees.  For example, at 9:00:00,
the hands are at 270 degrees OR 90 degrees, depending on how you
want to look at it.)

In general, for any angle, one of the times when the hands
will form this angle is:

a/330  hours, where a is the angle expressed in deg.

* Note that the answers will be in decimal forms, which can be
translated into the familiar hour-minute-seconds format.
Also note that this is only one of the 22 other times at which
the hands will also form the given angle.

For example, if the given angle is 60 degrees, then

60/330  = 0.1818...hours  = 0 hours 10 minutes 55 seconds.

Therefore, one of the times at which the hands will form a
60-degree angle is at 12:10:55 (0 hours 10 minutes 55 seconds
from 12:00:00).

---------------------------------------------------------------

The following table lists two of the angles and some of the
times at which the hands of the clock forms these angles:

Table 5.

Angle (deg.)  Times    Num. hours from 12:00:00
-----------------------------------------------------------------
90 = 360/4  12:16:22  0.27 hrs  = 90/330
12:49:05  0.82 hrs  = 90/330 * 3   (3 = 4 - 1)
01:21:49  1.36 hrs  = 90/330 * 5   (5 = 4 + 1)
01:54:33  1.91 hrs  = 90/330 * 7   (7 = 8 - 1)
02:27:16  2.45 hrs  = 90/330 * 9   (9 = 8 + 1)
03:00:00  3.00 hrs  = 90/330 * 11  (11 = 12 - 1)
03:32:44  3.55 hrs  = 90/330 * 13  (13 = 12 + 1)...

45 = 360/8  12:08:11  0.14 hrs  = 45/330
12:57:16  0.95 hrs  = 45/330 * 7   (7 = 8 - 1)
01:13:38  1.23 hrs  = 45/330 * 9   (9 = 8 + 1)
02:02:44  2.05 hrs  = 45/330 * 15  (15 = 16 - 1)
02:19:05  2.32 hrs  = 45/330 * 17  (17 = 16 + 1)
03:08:11  3.14 hrs  = 45/330 * 23  (23 = 24 - 1)
03:24:33  3.41 hrs  = 45/330 * 25  (25 = 24 + 1)...

There is obviously a pattern present.  It can be concluded from
studying the table that given the angle, all of the times at which
the hands of the clock forms the given angle can be determined.
The times seems to consistent to the pattern:

a/330 * (1 * 360/a  - 1)
a/330 * (1 * 360/a  + 1)
a/330 * (2 * 360/a  - 1)
a/330 * (2 * 360/a  + 1)
a/330 * (3 * 360/a  - 1)
a/330 * (3 * 360/a  + 1)...   where a is the angle in degrees.

Therefore, given the angle a, the times can be calculated by
the following formulae:

(360n - a)/330     and    (360n + a)/330

By substituting n in the formulae with the integers 1 through 11,
all 22 different times at which the hands of the clock forms the
given angle can be obtained.

Example:

Angle(a)  n  (360n-a)/330  time     (360n+a)/330  time
-----------------------------------------------------------------
90      1   0.82 hrs = 12:49:05    1.36 hrs = 01:21:49
2   1.91 hrs = 01:54:33    2.45 hrs = 02:27:16
3   3.00 hrs = 03:00:00    3.55 hrs = 03:32:44 ...
10  10.64 hrs = 10:38:11   11.18 hrs = 11:10:55
11  11.73 hrs = 11:43:38   12.27 hrs = 12:16:22

---------------------------------------------------------------

Using the method shown in Table 3, it can be shown that the
distance between the hands at 3:20:00 is the absolute value of:

[ 3(5) + 20/12 + 0/720 ]  -  [ 20 + 0/60 ]
=  [ 3(5) + 20/12 ]  -  [ 20 ]
=  3.33 units

(In comparison with the second equation in Table 3, the equation
[ 3(5) + 20/12 ]  -  [ 20 ]
has an added term: 3(5).  This is because the hour in Table 3
is 12 whereas the hour above is 3.  When the hour hand is at
12, it is 0 units from the origin, whereas when the hour hand
is at 3, it is 15 units from the origin (recall that the hour
hand moves 5 units every hour). )

We have just established that the hands are 3.33 units apart
at 3:20:00, now we shall find the angle.  Recall that 60 units,
is 1 revolution, or 360 degrees, around the clock. Therefore,
1 unit is equivalent to 6 degrees.  Since the hands are 3.33 units
apart at 3:20:00, it follows that the angle formed is
3.33 * 6 = 20 degrees.

In general, given the time in the format h:m:s (where h is
the hour, m the minutes, and s the seconds), the angle formed
by its hands is:

6 ABS[(5h + m/12 + s/720)  -  (m + s/60)]  degrees.

For example, at precisely 2:43:38, the angle formed
by the hands is:

6 ABS[(5(2) + 43/12 + 38/720)  -  (43 + 38/60)]
=  6(30)
=  180  degrees.

---------------------------------------------------------------

The following is a simple program I've written using Microsoft
QBASIC (which comes with Microsoft DOS).  If given a time, the
program can calculate the angle formed by the hands of the clock
at this time.  If given an angle, it generates a list of the 22
different times at which the hands will form the given angle.

-----------------------------------------------------------------
REM  Programmed by Kent Cheung 01/14/96

CLS
INPUT "Find (T)imes/(A)ngle "; R\$

IF UCASE\$(R\$)="A" THEN
CLS
PRINT "INPUT hh:mm:ss  °°:°°:°°"
LOCATE 1, 17
DO
k\$=INKEY\$
IF k\$<>"" THEN
PRINT k\$;
t\$=t\$+k\$: c=c+1: n=n+1
IF n=6 THEN EXIT DO
IF c>=2 THEN PRINT ":"; : c=0
END IF
LOOP
h=VAL(LEFT\$(t\$,2)): m=VAL(MID\$(t\$,3,2)): s=VAL(RIGHT\$(t\$,2))
a=6*ABS((5*h+m/12+s/720)-(m+s/60))
PRINT: PRINT"Angle formed by the hands at this time is";a;"deg."
END
END IF

IF UCASE\$(R\$) = "T" THEN
CLS
INPUT "Input angle: ", a
Format\$ = "\\:\\:\\  \\:\\:\\"
PRINT

FOR n=1 TO 11
t=(360*n-a)/330
GOSUB GetTime
h1\$=hr\$: m1\$=min\$: s1\$=sec\$
t=(360*n+a)/330
GOSUB GetTime
h2\$=hr\$: m2\$=min\$: s2\$=sec\$

PRINT USING Format\$; h1\$; m1\$; s1\$; h2\$; m2\$; s2\$
NEXT n
END
END IF

GetTime:
hour=INT(t)
minute=(t-INT(t))*60
second=(minute-INT(minute))*60
IF hour=0 THEN hr\$="12"
IF hour<>0 AND hour<10 THEN hr\$="0"+RIGHT\$(STR\$(hour),1)
IF hour>=10 THEN hr\$=RIGHT\$(STR\$(hour),2)
IF INT(minute)<10 THEN min\$="0"+RIGHT\$(STR\$(INT(minute)),1)
IF INT(minute)>=10 THEN min\$=RIGHT\$(STR\$(INT(minute)),2)
IF second<10 THEN sec\$="0"+RIGHT\$(STR\$(CINT(second)),1)
IF second>=10 THEN sec\$=RIGHT\$(STR\$(CINT(second)),2)
RETURN
-----------------------------------------------------------------

From: ruth@mathforum.org (Ruth Carver)
Rosie Twomey, Steph Miesnik, Courtney Piper

On a cursory examination of this problem, we thought, "This looks
simple de dimple."  There are 360 degrees in a circle and there
are sixty minutes so the angle formed by the radii of two adjacent
angles is 6 degrees.

However, after careful consideration, we realized that the hour
hand changes five times during each hour (it changes to each of
the minutes between the hour currently being experienced and the
soon-to-come hour). Since there are sixty minutes in each hour, we
divided sixty minutes by the five times the hour hand changes.
Thus, the hour hand moves six degrees (one minute) every twelve
minutes.

We realized that when you are solving for the angle, you have to
solve for the angle as well as 360 degrees minus the angle because
the formula looks at the angle clockwise but not if the angle is
faced counter-clockwise. This will be important to remember as you
try to understand why we solved for 90 degrees and 270 degrees for
question number one, and 45 degrees and 315 degrees for question
number two. Also if the hour you get for an answer is greater then
12, then subtract 12. If the minute you get for an answer is
greater then 60, then subtract 60.

Since each angle formed by the intersection of thte radii of
consecutive minutes is 6 degrees and you are trying to discover
the necessary number of minutes spaces between the hour hand and
the minute hand to form the given angle, you divide the angle by
6 degrees. In question number one for example, 90 degrees divided
by 6 equals 15. Thus, you should solve for the hour using 90
degrees and 270 degrees for "a" and every fifteen minutes for "m".
You should then have 8 different equations - four (:00, :15, :30,
:45) for 90 degrees and four (:00, :15, :30, :45) for 270 degrees.

1. The hour and minute hand form a 90 degree angle at two times
every 12 hours.  These times are 9:00 and 3:00.

The formula we used was:  (h=hours, m=minutes, a=angle)

h= (2a + 13m)/60
h= (2(90)+ 13(0))/60
h=(180+0)/60
h=180/60
h=3  m=:00 time=3:00

h=(2a +13m)/60
h= (2(270) +13(0))/60
h=(540+0)/60
h=540/60
h=9 m=:00 time =9:00

We also did this six more times for the different minutes
and angles but these were the only ones that came out.

2. We also tried to find the hour in an equation containing the
angle and the minutes, so we divided 45 by 6.  The answer was 7.5.
Then we made sixteen equations - eight (:075, :15, :225, :30,
:375, :45, :525, :00) for 45 degrees and eight (:075, :15, :225,
:30, .375 :45, :525, :00) for 270 degrees). The hands of a clock
form 45 degree angles only at 8:30 and 5:30.  A generic formula to
solve for an hour if you have the angle would be:

h= (2a +13(a/6))/60

(If you consider the fact that in order to get from one minute to
the next you must pass through each of the six degrees for a
quintessence of a second, then the clock does form a 45 and 90
degree angle twice every hour but for much less then a second)

3. The formula we got was 6[ | 5h-m | -m/12]=a  Also if the angle
answer is greater then 180 degrees, then subtract 180 from the

6[ | 5(3)-20| -20/12]=a
6[ | 15-20 | -20/12]=a
6[ | -5 | -20/12]=a
6[ 5-20/12]=a
6 [ 40/12]=a
240/12=a
20 degrees=a

I didn't know how to solve for the seconds.  I was confused
because three distinct lines can not form just two angles.
They form three.

From: ruth@mathforum.org (Ruth Carver)
Katrina Myers and Karen Wing

1. To start this project, we looked at the units of time which are
at a 90 degree angle to each other.  We found these
relationships: 12 is 90 degrees from 3, 1 from 4, 2 from 5,
3 from 6, 4 from 7, 5 from 8, 6 from 9, 7 from 10, 8 from 11,
9 from 12, 10 from 1, and 11 from 2.

We wrote all the possibilities down for the 12, 1 and 2 o'clock
hours. From this, we concluded that the three hands of a clock
form 90 degree angles 120 times.  We found that 90 degree angles
will form when any hand is three hour units away from one or both
of the others.

ex.12:00:151:00:202:00:25
12:00:451:00:502:00:55
12:15:001:20:002:25:00
12:15:301:20:352:25:40
12:15:451:20:502:25:55
12:30:151:35:202:40:25
12:30:451:35:502:40:55
12:45:001:50:002:55:00
12:45:151:50:202:55:25
12:45:301:50:352:55:40

2. We drew a scale clock and measured a 45 degree angle from 12
with a protractor.  The time was 12:07:30.  We drew 45 degree
angles from other time units and decided to find out if we
could come up with the answers mathematically.  We thought a
proportion would be the easiest method and established the
relationship as:  the measure of the angle to 360 degrees
equals the minute unit distance from the hour unit to 60
minutes.

For example: 90 degrees is to 360 degrees as 15 minutes (from
12) is to 60.

We used our method to prove that 00:07:30 is 45 degrees from 12.

5/360=x/60 2700=360x7.5=x
7.5 converts to 7 minutes and 30 seconds

Our equation, therefore, is: angle m/360=minute time/60

3. We decided to put our equation to the test.  To find what angle
the hands of a clock would be at 3:20, we plugged it into the
equation.

m/360=5/6060m=1800m=30

The measure of the angle of the hands on a clock at 3:20 is 30
degrees. We checked our work with a protractor and found our

From: ruth@mathforum.org (Ruth Carver)
Jackie Benn and Shannon Firth

1) We found that the hands of the clock will make a 90 degree
angle at 48 different times.   Since we knew that a circle
(the clock) is 360 degrees and there are 12 numbers of the
face of the clock evenly spaced, we divided 360 by 12 and got
30 degrees for the distance between every number.  That would
mean that when the hour hand and minute hand were three spaces
away from eachother (with the second hand overlapping one of
the hands), they would form a 90 degree angle. These are the
times we found:

12:15.15, 12:15.00, 12:45.45, 12:45.00, 1:20.20, 1:20.05,
1:50.50, 1:50.05, 2:25.25, 2:25.10, 2:55.55, 2:55.10,
3:30.30, 3:30.15, 3:00.00, 3:00.15, 4:05.20, 4:05.05,
4:35.20, 4:35.35, 5:10.25, 5:10.10, 5:40.40, 5:40.25,
6:15.15, 6:15.30, 6:45.45, 6:45.30, 7:20.20, 7:20.35,
7:50.35, 7:50.50, 8:25.25, 8:25.40, 8:55.55, 8:55.40,
9:30.30, 9:30.45, 9:00.00, 9:00.45, 10:05.05, 10:05.50,
10:35.50, 10:35.35, 11:10.10, 11:10.55, 11:40.40, 11:40.55.

2) A time when the hands of a clock will make a 45 degree angle
is when the hour hand is on 1, and the minute hand is between
the numbers 12 and 13.  Since we knew that every three numbers
form a 90 degree angle, that a 45 degree angle would be exactly
half of that (1 and a half numbers away).  This can apply to
all numbers, but you have to remember that the minute hand must
be exactly in between the second and third dash mark between the
numbers.

3) Knowing that angle between each number on the face of the clock
is 30 degrees, and that when the clock reads 3:20 one hand will
point to 3, the other to 4 (which represents 20), the hands will
make a 30 degree angle. Our method can be applied to any given
time.  All we did was count the number of spaces between the two
numbers and multiply that by 30.

From: ruth@mathforum.org (Ruth Carver)
Susan McGowan
Mt. St. Joseph Academy, Flourtown PA
Sophomore

The times when a right angle are formed by the hands of a clock
are:

12:16.0             6:16.16
12:16.16            6:16.36
12:49.4             6:49.34
12:49.49            6:49.49
1:22.7              7:22.22
1:22.22             7:22.37
1:54.9              7:54.40
1:54.54             7:54.54
2:27.12             8:27.27
2:27.27             8:27.42
3:00.0              9:00.0
3:00.15             9:00.45
3:33.18             9:33.33
3:33.33             9:33.48
4:05.5             10:05.5
4:05.21            10:05.51
4:38.23            10:38.38
4:38.38            10:38.53
5:11.11            11:11.11
5:11.26            11:11.56
5:44.29            11:44.44
5:44.44            11:44.58

In order to have a right angle, the hour and minute hands must be
16 second dashes away from each other. The second hand must then
line up with either the minute or hour hand. When it rests on the
minute hand, the seconds number will be the same as the minute.
When it rests on the hour hand, it will line up with one of the
five dashes within he hour markers, depending on how late in the
hour it is. A right angle will appear 88 times a day. I have only
written 44 because each of these symbolizes the a.m and p.m.
times.

A time at which a 45 degree angle is produced is 12:09.  In order
to achieve a 45 degree angle the hands must be 8 second dashes
away from each other.  In order to find any angle you use this
equation-1 second dash = 5.625 degrees.

The time 3:20 produces a 17 degree angle.  Use above generalized
equation.

From: ruth@mathforum.org (Ruth Carver)
Cindy Spering

1. During the course of a 24-hour period, the clocks hands form
right angles. These are as follows:

9:00   (and, of course, these are the same for a.m. and p.m.)
9:32
10:07   I found this out by taking and old alarm clock
10:40   and turning the hands. Every time a 90 degree angle
11:13   occurred, I'd record it.  Counting the number of times
11:44   this occurred, I arrived at the conclusion that the clock
12:16   forms 44 right angles in a typical day.
12:49
1:21
1:54
2:27
3:00
3:35
4:05
4:40
5:11
5:44
6:16
6:50
7:21
7:55
8:26

2. Clocks form a 45 degree angle at 8:35. To find any given angle,
(of course, this may sound a bit uncomputerish) you can do the
conventional method of cutting a piece of paper in the desired
angle measurement, and moving the arms of the clock so they form
the desired angle.

3. At 3:20, the clock forms a 245 degree angle. This was found by
dividing the 360-degree circle by 60, to find out how many degrees
of a circle are represented by one minute mark(6.) There are 4 of
these marks between the hour and minute hand of 3:20. This can
solve for any angle if you use the number of minute marks times 6,
and the angle will be given.

(Much to my alarm, I discovered that you can not use a digital watch
when calculating the angle formed by a clock! }:^)   .)

From: ruth@mathforum.org (Ruth Carver)
Omua Ahonkhai

2. The hands of a clock will form a 45-degree angle at approximately
12:09.

I used a very simple method to find this time.  First, I looked at
the properties of perpendicular lines and found two lines that form
any 90-degree angle, always form four 90-degree angles.  The numbers
on a clock form a circle: to divide a circle into four 90-degree
angles the circle would have to be divided into four equal parts.
So I imagined two lines that would divide a clock into four equal
parts, one line passing through the twelfth and sixth hour, and one
passing through the ninth and third hour.  I now had four 90-degree
angles   Each contained a period of 15 min. To form a 45-degree
angle I had to find a time including a period of 7.5 min., about
eight minutes, between the two hand of the clock.  I chose the
time 12:09.

A general method to find a time at which the hands of the clock
form any given angle is to use the following formula: let "m"= the
amount of periods of a minute between the two hands (rounded to the
nearest minute) 6-degrees(m) After finding that a 90-degree angle
is formed with 15 min. between the two hands, I set up a proportion
of 90-degrees /15 min.  = x-degrees/1min. x = 6-degrees. So 6-degrees
multiplied by the amount of periods of a minute between the two hands
of a clock will equal the degree of an angle formed by the hands of a clock.

3. Using this formula I knew that the hand of a clock form an
18-degree, 6-degrees(3), angle.

From: ruth@mathforum.org (Ruth Carver)
Jennifer Keeney and Kathleen Wuerth

1. There are 30 degree angles between any two numbers on the clock.
We marked off all the possible times at which the minute and
hour hands create a 90 degree angle by always placing the hands
3 numbers apart.  We did this all around the clock and found
17 different times where a 90 degree angle is formed. They are:

12:15, 1:20, 2:25, 2:50, 3:00, 3:30, 4:05, 5:10, 6:15, 7:20, 7:50,
8:25, 8:55, 9:00, 9:30, 10:05, and 11:10.

We did not know what seconds had to do with it.

2. To find a 45 degree angle on the clock, first we divided the clock
into 4 equal parts, using a vertical line through the twelve and
the six, and a line perpendicular to it , running through the nine
and the three.  Each of these sections has a time span of fifteen
minutes. So if we divide each of the four sections in half, we
will have eight equal sections, each with an angle of fourty-five
degrees, and a time span of seven and one-half minutes.

An example of this is when the hour hand lies on the twelve, and the
minute hand lies between the seven- and eight- minute marks.

3. At three twenty, the hands form a thirty degree angle. A general
method of finding any angle formed the hands on a clock wasn't
that hard to find. Since there are three hundred and sixty
degrees in a clock's face, and there are twelve number marks on
the clock, we divided three hundred and sixty by twelve. That
comes out to thirty degrees between each number mark. There are
sixty minute marks on the face of a clock, so when we divide
three hundred sixty degrees by sixty, we find out that there are
six degrees between each minute mark. Thanks for reading. Peace!

From: ruth@mathforum.org (Ruth Carver)
Kristy Giballa

The first question was a little too complicated for me so here are
the answers to questions two and three.

Before I give the specific answers I will generalize my method
because I used the same one for both questions. A circle has 360
degrees and a clock is... a circle! So if you use this information
you can find that between each number there are 30 degrees, each
minute has 6 degrees and each second has 1 degree.

2. A time at which the hands of the clock make a 45 degree angle
is 12:07:30.

3. The angle that the hands of a clock form at 3:20 is 18 degrees.
This problem was not as easy as I though it was.  I used a
watch as an aid and I realized that excluding on the hour the
little hand does not stay on the number(at 3:15 the small hand
is slightly after the three)  By turning the watch I found the
difference and fixed it.

From: ruth@mathforum.org (Ruth Carver)
Jill Sommer

After a lot of time staring at my watch and thinking, I came up with these answers to the January Problem of the Month.

1) At first, I could only think of a few 90 degree angles formed
by the hands of a clock.  I thought of 3:00 and 3:30, but then
I looked at the clock and realized that there had to be many 90
degree angles formed.

After a while, I found 48 times when this would happen. The times
are:

1:20:05and20, 1:50:05and50, 2:25:10and25, 2:55:55and10,
3:00:00and15, 3:30:15and30, 4:05:20and05, 4:35:20and35,
5:10:25and10, 5:40:25and40, 6:15:30and15, 6:45:30and45,
7:20:35and20, 7:50:35and50, 8:25:25and40, 8:55:40and55,
9:00:00and45, 9:30:30and45, 10:05:05and50, 10:35:35and50,
11:10:10and55, 11:40:40and55, 12:15:00and15, and 12:45:00and45.

Then I thought that if the hour and minute hands were together and
the second hand was a quarter of the way around the clock from
there, they would form a 90 degree angle.  Then., there would be
more times such as 12:00:15and45.  This would happen 118 other
times, but I didn't even want to begin writing all these, sorry.

A 90 degree angle is formed whenever a hand of the clock is
fifteen seconds or minutes away from another hand.

2) Since the space between every dash is 360/60, each one measures
6 degrees. In order to make a 45 degree angle, the second hand
would have to be 7.5 seconds away from another hand, or the
minutes hand would have to be 7.5 minutes away from another
hand. For example: 6:30:37.5, or 1:07:30 and so on.

angle measure/ 6= the space in minutes or seconds that
separate two hands.

3) Since the 3 and 4 on a clock are 5 spaces away, you can
multiply 5 by 6 and see that the hour hands at 3:20 will form a
30 degree angle.  To get the angle formed from any time, count
the seconds between the two hands and multiply by 6.

From: ruth@mathforum.org (Ruth Carver)
Michele Weiss

This problem took me weeks to figure out.  I'll first tell you
how I started out and then I'll give you the answers.

First, I made a lot of equivalencies that were sometimes not
needed but helped me in generalizing my equations. Here they are:

Minute hand moves 12 min.    =  Hour hand moves 1 min.
Minute hand moves 72 degrees =  Hour hand moves 6 degrees
Minute hand moves 6 degrees  =  Hour hand moves 1/2 degree
Minute hand moves 1 min.     =  Hour hand moves 1/12 min.
Minute hand moves 60 sec.    =  Hour hand moves 5 sec.
Minute hand moves 1 min.     =  Hour hand moves 1/2 degree

5 min. = 30 degrees
1 min. = 6 degrees
60 sec. = 6 degrees
1 sec. = .1 degree

From these I figured out that every minute change on the minute
hand = 6 degrees and at the same time, the hour hand moves 1/12
of a "minute" or 1/2 of a degree.  I came up with some examples
of this:

12:01 = + 6 - 1/2   = 5 1/2 degree angle
12:02 = +12 - 1     = 11 degree angle
12:03 = +18 - 1 1/2 = 16 1/2 degree angle
12:04 = +24 - 2     = 22 degree angle

Because of this, I came up with this equation for the 90 degree
angles in the 12 o'clock hour:

90 = 6x - 1/2x
270 = 6x - 1/2x

-where-

x = the minutes after the hour

When using these equations, you will always get a repeating decimal.
Change your decimal to a fraction and then convert into seconds by
multiplying by 60.

After this, I realized that this was only an equation for the
12 o'clock hour.  I came up with another equation to figure out
what kind of angle you have at a given time and vice-versa.

h + (n*30) = 6 x - 1/2x
-where-
x = minutes after the hour
n = hour
h = degrees of the angle

* exception:  If it is the 12 o'clock hour that you are solving for,
n = 0 as the first hour of the day

* NOTE - In the equation, eventually (n*30) will get bigger than a
360 degree angle, or a circle.  When this happens, subtract
60 from the  number you get for the answer.

Now on to the questions:

1. The clock will make a right angle 22 times.  Every hour has
2 right angles in it except 2 o'clock and 8 o'clock.

These are the times at which the right angle will occur:

HOUR1st Right Angle  2nd Right Angle

12:0012 : 16 : 22   12 : 49 : 05
1:001  : 21 : 49    1 : 54 : 33
2:002  : 27 : 16   no second one
3:003  : 00    3 : 32 : 44
4:004  : 05 : 27    4 : 38 : 11
5:005  : 10 : 54    5 : 43 : 38
6:006  : 16 : 22    6 : 49 : 05
7:007  : 21 : 49    7 : 54 : 33
8:008  : 27 : 16   no second one
9:009  : 00    9      : 32 : 44
10:0010 : 05 : 27   10 : 38 : 11
11:0011 : 10 : 54   11 : 43 : 38

2. I chose to use the 12 o'clock hour for my equation.

h + (n*30) = 6x - 1/2x
45 + (0*30) = 6x - 1/2x
45 = 6x - 1/2x
45 = 5.5x
8.1818.....= x
8 2/11 = x
8 : 11 = x
The time is 12 : 08 : 11.

3.  I use my equation and solve for the unknown variable.

h + (n*30) = 6x - 1/2x
h + (3*30) = 6(20) - 1/2(20)
h + 90 = 120 - 10
h + 90 = 110
h = 20
The angle will be a 20 degree angle.

From: ruth@mathforum.org (Ruth Carver)
Katie Walder, Lindsay Parsons, and Susan Tull

1. The 2 most obvious answers are 3:00 and 9:00, morning or night.
This is true, but there are many more.  Take 3:00 for example, for
every time you add one hour to it, add 5 minutes.  This will
always form a right angle.(It would be at 4:05, 5:10, 6:15 and so
on.0 This same idea would also be true for 9:00.(It would be at
10:05, 11:10, !2:15 and so on).  This would give us 12 right
angles for each, twice a day, so there would be a total of 48
definite right angles during a day using only the minute and hour
hands.

When using the second and minute hands, the second hand will make
a right angle twice a minute, 120 times an hour. (These times
will be when the second hand is in the spot 15 minute spaces away
from where the minute hand is.)  There must also be 120 times an
hour when the second and hour hands form right angles.  When using
all of the hands there are approximately 5,800 times that right
angles are made on a clock.

2. A 45 degree angle will be formed 11,600 times a day.  The
reason for this being is that for every right angle formed, there
will be 2 45 degree angles.  Therefore you multiply your total
number of right angles by 2.

A formula could be made by the idea that a circle is 360 degrees.
Every hour is 30 degrees, every minute is 6. Figure out the angle
by what hour and what minute the hands are at.

3. At 3:20 the hands of the clock will form a 30 degree angle.
This will occur because there are 360 degrees in a circle.  When
the circle is split into the 12 hour spaces each is 30 degrees.
Also each minute space is 6 degrees.  Knowing this, you could find
any angle formed by the hands of a clock.  When it is 3:20 AM or
PM one hand will be on the "3" and one on the "4" thus creating a
30 degree angle.

From: ruth@mathforum.org (Ruth Carver)
Kelly Larkin, Liz Croney, and Annie McIntyre

The hands of a clock will form right angles at: 12:15, 1:20, 2:25,
3:30, 4:35, 5:40, 6:45, 7:50, 8:55, 9:00, 10:05, 11:10.

We then realized that you can switch the hour and minute hands of
these times and you would get new times, but still 90 degree
angles.  These times are: 3:00, 4:05, 5:10, 6:15, 7:20, 8:25,
9:30, 10:35, 11:40, 12:45, 1:50, 2:55.

There are many other ones that will follow the pattern above, "add an
hour and 5-minute method." The hands of a clock will form 45-degree
angles at 12:07 and 30 seconds. While trying to find this out, we
discovered that each minute is a 6-degree angle, every five minutes
is a 30-degree angle, and every 7.5 minutes is a 45-degree angle, and
so on.

At 3:20, the hands of a clock will form a 30-degree angle according
to our method stated in our second paragraph.  Our equation to find
the angle formed for a time is: the amount of minutes in between the
two hands multiplied by six.  Our equation to find the time for a
given angle is: the degrees given divided by six.

From: ssusd2@owens.ridgecrest.ca.us
Cassie Gorish
School: Murray Junior High

The times are:

3:00 + 00
3:32 + 44
4:05 + 27
4:38 + 10
5:10 + 54
5:43 + 38
6:16 + 22
6:49 + 05
7:21 + 49
7:54 + 32
8:27 + 16
9:00 + 00
9:32 + 43
10:05 + 27
10:38 + 10
11:10 + 54
11:43 + 38
12:16 + 21
12:49 + 09
1:21 + 48
1:54 + 32
2:27 + 16

To the first part of the second question:

At 3:08 + 11 sec.

To the second part:

First:

Solve an equation:

.1 degrees=minute hand speed
.0083 degrees=hour hand speed
T = time
S = seconds
n degrees = number of degrees to find.

_.1d_  x  T =  _.0083d_ x  T + n degrees
s                                  s

Second:

Set a clock to 12:00 for 12:00 + 3927.2 sec. (where the hands
will be together again).  Add or subtract n seconds.

To the first part of the third question:

The angle is 20 degrees.

To the second part:

First: Find the seconds (T) between the time given and the
nearest hour/minute hand position where they are
together.

Second: Solve for T:

.0916T = n degrees

T = seconds measured above.

From: megan@fermat.whitman.edu
Megan Guichard
Pioneer Middle School, Walla Walla, WA

Part One:

During a twelve-hour period, there are 22 different times when the
hands of a clock form right angles (44 times during a full day -
3:00 occurs once in the morning and once in the afternoon, as do
all the times I've listed), which I have listed below, in both
rounded and un-rounded form (it turns out to be quite easy to find
exact times using the formula I made) and grouped by the hour (in
the form hours:minutes:seconds):

Time Group:            Unrounded Time:                Rounded Time:
12:00:00 to 12:59:59   12:16:21+9/11  12:49:05+5/11   12:16:22  2:49:05
1:00:00 to 1:59:59     1:21:49+1/11   1:54:32+8/11    1:21:49  1:54:33
2:00:00 to 2:59:59     2:27:16+4/11                   2:27:16
3:00:00 to 3:59:59     3:00           3:32:43+7/11    3:00:00  3:32:44
4:00:00 to 4:59:59     4:05:27+3/11   4:38:10+10/11   4:05:27  4:38:11
5:00:00 to 5:59:59     5:10:54+6/11   5:43:38+2/11    5:10:55  5:43:38
6:00:00 to 6:59:59     6:16:21+9/11   6:49:05+5/11    6:16:22  6:49:05
7:00:00 to 7:59:59     7:21:49+1/11   7:54:32+8/11    7:21:49  7:54:33
8:00:00 to 8:59:59     8:27:16+4/11                   8:27:16
9:00:00 to 9:59:59     9:00           9:32:43+7/11    9:00:00  9:32:44
10:00:00 to 10:59:59   10:05:27+3/11  10:38:10+10/11  10:05:27 10:38:11
11:00:00 to 11:59:59   11:10:54+6/11  11:43:38+2/11   11:10:55 11:43:38

Explanation:

At any given time x:y, where x is the hour and y is the number of
minutes past the hour, the minute hand is 6*y degrees, clockwise,
away from the 12 (there are 360 degrees in a circle and 60 minutes
in an hour, so there must be 360/60 or 6 degrees between the marks
for each minute). Likewise, the hour hand must be 30*(y/60 + x) or
(1/2*y + 30*x) degrees, clockwise, away from the 12. (At the time
x:y, the hour hand will be y/60 of the way between the x marking
and the (x+1) marking, giving you, since there are 360/12 or 30
degrees between the hour markings, 30*(x + y/60) degrees beyond
the 12.)

Given this, the difference in degrees between the hour hand
and the minute hand must be the positive difference between the
degrees past 12 that the minute hand is and the degrees past 12
that the hour hand is, which can be written as

|6*y - (1/2*y + 30*x)| =
|6*y - 1/2*y - 30*x| =
|5.5*y - 30*x|,

where |a| is the absolute value of a.

I then used this formula to find all the times at which the two hands
of a clock form a right angle by putting in all the integers from
0 to 11 for x (0 for 12, because it's easier to deal with 0 degrees
than 360 degrees) and solving the equations |5.5*y - 30*x| = 90 and
|5.5*y - 30*x| = 270 with each of these integers (when the two hands
form a 270-degree angle, they also form a (360-270) or 90-degree
angle), disregarding answers in which y  60, as there are only 60
minutes in an hour, and in which y < 0, because you can't have a
negative number of minutes.

Part Two:

One time at which the hands of a clock make a 45-degree angle is
4:30, which I found by using the formula |5.5*y - 30*x| = 45,
which I explained above, and putting 30 in for y (because half-
past any hour is a nice, round number). Solving this equation, I
got x = 4 or x = 7, but since the question only asks for one time,
I left off 7:30.

In general, as explained in Part 1, to find a time at which the
hands of a clock form a given angle a, you can use the formulas

a = |5.5*y - 30*x| and (360-a) = |5.5*y - 30*x|,

putting any integer from 1 to 12 in for x (0 to 11 if you're like
me and prefer to deal with 0 degrees rather than 360 degrees), as
there are quite a few solutions for any given angle a. The formula
(360-a) = |5.5*y - 30*x| is included because, as I mentioned
before, when the two hands of the clock form an angle of (360 - a)
degrees, they will also form an angle of a degrees, meaning that
solutions to both equations will work.

Part Three:

At 3:20, the hour hand and the minute hand of a clock form a
20-degree angle, which I found using the formula explained in
Part 1.

In general, as I explained in Part 1, to find the angle, a, that
the hands of a clock form at x:y (x, again, being the hour and y
the number of minutes past the hour), you can use the formula

|5.5*y - 30*x| = a.

Again, the angle formed by the two hands will be either a or
(360 - a), as the hands will form both of these angles.

From: ranierm@firnvx.firn.edu
Jacqueline Baras
School: Bay Point Middle School, St. Petersburg, FL

How to get the solution: (NOTE: these formulas are in degrees, not
minutes.)

Because I didn't see any obvious pattern, I first found the ratios
between the rate of movement of the hour-hand compared to the rate
of the minute-hand and stated additional information:

- every 6 degrees (a minute) the minute hand moves, the
hour hand moves 1/2 degrees 6 : 1/2

- every five minutes is equal to 30 degrees.

I used an actual clock at first for the purpose of visualizing how
a clock works, to get a good idea about it. Also, I logically
thought to use the idea of the Parts Theorem on the circle to find
the angle degrees. An imaginary ray was "drawn" from the center of
the circle/clock to a point on the 60 minute mark of a clock
(where the "12" would be). This would be the ray that correlated
with the ray of the minute-hand and the ray of the
hour-hand, creating a minute-hand angle and an hour-hand angle.
So, if the measure of the minute-hand and hour-hand angles were
known, one could easily find the measure of the angle made by both
hands by subtraction.

Because of the immense amount of times/angles to find out, and
still not seeing any formula,  I used a spreadsheet for each hour.
Each spreadsheet contained the degree mark/angle measure of the
hour hand (at a rate of  .5 degrees per minute) and the minute
hand (at a rate of 6 degrees per minute) for each minute. For
example, at 4:20, the minute hand was on the 120 degree mark and
the hour hand was on the 130 degree mark. Then, from looking at
the actual clock with some common reasoning, I used simple
subtraction and addition formulas to find the angle measure formed
by both hands (formulas are below under sections No. 2 and No. 3).
After making the countless spreadsheets, I eventually found the
detailed formulas, understanding truly how the clock operates, as
well as the fact that two times, even a second apart, make two
very different angles.

1) From the spreadsheets, I discovered 22 different times where
the clock hands made a right angle (90 degrees). Surprisingly,
these times were all 32:44 to 33:44 apart. Perhaps this happens
because every 33 1/2 minutes or so, the ratios of the rates of the
hour and minute hand come together to make the hands 90 degrees,
or 15 minutes apart. Here are the following times, determined to
the nearest second:

3:00:00    5:10:54    7:21:49    9:32:44   11:43:38    1:54:33
3:32:44    5:43:38    7:54:33   10:05:27   12:16:22    2:27:16
4:05:27    6:16:22    8:27:16   10:38:11   12:49:05
4:38:11    6:49:05    9:00:00   11:10:54    1:21:49

2) An obvious time where the hands of the clock make a 45-degree
angle (7.5 minutes apart) is 7:30. Because of the observances in
the above problem, I realized that to find a time at which the
hands of a clock form any given angle, I would have to subtract
the degrees of the larger hand-angle (the angle consisting of the
min or hr. hand {a ray} and the imaginary ray from the center of
the circle to the 60 minute point of a clock) from the degrees of
the smaller hand-angle. From the following information, I could
find a time:

* The minute hand moves 6 degrees every minute
* The hour hand moves 1/2 degree every minute
* There are 360 degrees in a circle
* Between every five minutes (in other words, between every
"number" on the clock) is 30 degrees
* There are 60 seconds in a minute

In addition, because the question said 'find A time where the
hands of a clock form any given angle', it is allowing us (and it
is necessary to) know at least the hour in which the given angle
will occur  (From the spreadsheets, I found out that between every
hour, there is at least one angle of every measure(0-180). If you
know the hour number then you can find the angle.

Okay, now let's find a time, X  (hour) : Y (min) : Z (sec) that
has a measure of 130 degrees. Let's look for one between 8:00 and
8:59. We know that the hour hand begins on  30X (between every
"number" is 30 degrees) each hour. We have to remember that the
hour hand is moving at .5 degrees every minute, so at Y minutes,
the hour hand has moved .5Y degrees from its original point. So,
the hour hand is at 30X + .5Y degrees. The minute hand, moving
6 degrees every minute, is at 6Y degrees. (We will talk about
seconds later) Now we can subtract the larger hand-angle from the
smaller hand-angle (6Y- 30X - .5Y  or  30X + .5Y - 6Y) which is
simplified to  5.5Y - 30 X or 30X- 5.5Y. Because we have to find
an angle with 130 degrees, use the equations: 5.5Y - 30X= 130 or
30X- 5.5Y=130 (remember to substitute 8 for X). If calculated, you
can see that the second one works for the time to be 8:20:00.

If Y is a non-whole number, then take that decimal and convert it
into seconds by the proportion: sec/60 = min/100. Back in review,
here are the formulas.

a) Let X = hour, Y = min, Z = sec, and A = angle measure

b) Knowing X and A, substitute those numbers in the equations
5.5Y - 30X  = A (minute angle > hour angle)
30X   - 5.5Y = A (hour angle>minute angle)
* if you don't know if the min ang>hr. ang, try both equations

** When X= 12, use the equations
5.5Y = A (min angle> hour angle) or
360 - 5.5Y = A (hour angle> minute angle)
Why? There is no need for a 30X since the hour hand was originally
on the 0/60 degree mark.

c) You have found Y if Y<60 and is a positive number.

d) If Y is a non-whole number, then take that decimal and convert it into seconds by the proportion: sec/60 = min/100.

3) The hands of a clock will form a 20 degree angle at 3:20. We
can show how to find the angle formed by the hands of a clock at
any given time using the formulas below which are similar to #2's
formulas:

a) Let X = hour, Y = min, Z = sec, and A = angle measure

b) Convert seconds into minutes by using the proportion: sec/60 =
min/100. Add the decimal to Y.

c) Knowing X and Y, substitute those numbers in the equations
5.5Y - 30X = A (minute angle > hour angle)
*use this equation if 5.5Y>30X
30X - 5.5Y  = A  (hour angle>minute angle)
*use this equation if 30X>5.5Y
* When X= 12, use the equations
5.5Y = A (min angle> hour angle)
360 - 5.5Y = A (hour angle> minute angle)

d) If A>180, then use the expression 360 - A to get the REAL
angle degree measure. Why? There is no such thing as an angle
greater than 180 degrees. Because the angle is a central angle,
the formula was stating the degree measure of the other "part of
the 360 degree pie".

e) There may be times in which in an hour, 2 angles with the same
measures occur with the hour angle>minute angle, or vice versa,
such as 10:05:27 and 10:38:11. In this case, use these additional
formulas 360 - 5.5Y+ 30X = A or 360- 30X + 5.5Y = A.

** There are 22 times when the hands of a clock make a 45 degree angle.

From: ranierm@firnvx.firn.edu
Aaron Cropper, Melissa Crow, Alex Allred
Bay Point Middle School, St. Petersburg, FL,

1) Clocks And Right Angles:

Obviously, counting through and checking every time on a clock for
right angles would take a very long time and be unreliable. Thus,
we looked for a pattern. As we know, circles have 360 degrees in
them , and right angles have 90 degrees. From this we can derive
several key measurements about the amount of degrees in certain
amounts of times:

time      degrees hour hand moves   degrees minute hand moves
1 hour             30.00                  360.00
1 minute           00.50                   06.00
1 second           00.008333               00.1

Now, we'll pick a beginning time, say 3:00, because it is a known
right angle (there is a 15 minute difference using the minute
hand, which translates to 90 degrees).

The next step is to estimate the next right angle, which ends up
being around 3:30.  Since the hour hand has moved in the 30
minutes, you have to discover how much, in order to catch the two
hands up.  30 minutes (times) .5 degrees per minute is 15 degrees.
This means that the hour hand has moved 15 degrees in 30 minutes.

In order to make up for that, the minute hand has to move a
certain amount of time.  To figure that amount of time out, you
have to use the information in the chart above.  6 (the amount of
degrees the minute hand moves in  a minute) goes into 15 twice,
giving a quotient of 2 minutes, and leaving 3 degrees as a
remainder.  0.1 (the amount of degrees the minute hand moves in a
second) goes into 3 degrees, 30 times, leaving 2 minutes and 30
seconds.

You then add that to the original time of 3:30.  However, the hour
hand has moved MORE in that time.  So, you have to make up for
that.  To do so, you find out how much it moved in 2 minutes and
30 seconds.  You multiply 2 (times) 0.5 (from above chart), and
you add 30 (times) 0.0083333.  This will leave you with
approximately 1.2.  Divide that number by 0.1, because that is how
many degrees are in a second, for the minute hand, and that leaves
you with 12.  You then add 12 to the original time, 3:30.  The
problem would look like this:

3:30
2.30
+  .12
3:32.42 is the official right angle.

Using the above information, we discovered that there were 22
right angles occurring in a 12 hour cycle, approximately 1 every
33 minutes.  Using the method described above, you can get the
exact times of the right angles:

1:21.48
1:54.30
2:27.15
3:00.00
3:32.42
4:05.27
4:37.65
5:10.54
5:43.36
6:16.21
6:49.04
7:21.48
7:54.30
8:27.15
9:00.00
9:32.42
10:05.27
10:37.65
12:16.21
12:49.04

2) 45 degree angles:

There are many times when the hands of a clock form 45 degree
angles, and here are two examples: 12:08.10  and 5:21.50.
The reason for this is similar to the explanations above, given
for 90 degree angles.  Obviously, you don't estimate the right
angles this time; you look for 45 degree angles or when the hands
are 7.5 minutes apart.  This formula can be used for any angle.

3) What angle do the hands form at 3:20?

They form a 20 degree angle.

Explanation:  (This formula can be used to discover any time's
angle) To figure out what angle a certain time creates,  do the
following: Discover what angle the minute hand is forming from the
12 O'clock mark. (6 degrees per minute, times however many
minutes)  Then discover what angle the hour hand is at, from the
12 O'clock mark. For example, we shall show what the measure is of
the angle formed by the hands of the clock at 4:20.  The hour
hand, if at exactly the 4 O'clock, would form a 120 degree angle.
But it is twenty minutes past four. Since the hour hand travels
10 degrees in 20 minutes(.5 degrees per minute times 20), add that
to 120 degrees and get 130 degrees. The minute hand would be a 120
degree angle, since it is at the 20 minute mark.  You then
subtract the smaller number from the bigger number to get the
amount of degrees between them. So, 4:20 would be a 10 degree
angle.

From: ranierm@firnvx.firn.edu
Amy Bonsey,   Kerega Melville, Jason Borowski
Bay Point Middle School, St. Petersburg, FL

1) 12:17    12)  6:17    This is a chart of the times that
2) 12:49    13)  6:49    make right angles.  They are as
3)  1:23    14)  7:23    best as we could judge by
4)  1:55    15)  7:55    taking a clock off the wall and
5)  2:28    16)  8:28    spinning it around and writing
6)  3:00    17)  9:00    down the times that are right
7)  3:33    18)  9:33    angles. Of course these times
8)  4:06    19) 10:06    would come up twice once
9)  4:39    20) 10:39    for A.M. and then again for
10) 5:12     21) 11:12    P.M..
11) 5:45     22) 11:45

* From 12:17 to 5:45 the times are the same as 6:17 to 11:45 in
the minute hands.  Look at the corresponding numbers that are
across from each other , such as 11 and 22,   4 and 5

* There are two times with right angles in every hour, except:
2 and 8 , which have only one because they are right before the
only two hours on the clock that are right angles, 3:00 and 9:00

* The time between each marked right angle time has an average of
32 minutes and it ranges from 27 to 24 minutes.12 hours times 60
for min. in hour.

12x 60=720  720 min. in 12 hours.  720/22= 32.72

Each second mark on the clock is 3 degrees because if one of the
hands was on the 6 and the other on the 9 you would have a
straight line that is 180 degrees.  So you take 180 /the number of
second marks in a half hour which is obviously is 30. (180/30)= 6.
You can find out any degrees just by counting the little second
marks as 6 degrees each.

2. You know that there are 360 degrees in a circle and 60 minutes
marks on a clock so you know that between each mark there are
6 degrees . Since 15, 6 degree angles make up 90 degrees, you know
that 7.5, 6 degree angles make up 45 degrees( which is half).
Therefore, there has to be 7.5 minute marks between the minute
hand and the hour hand.  Also, I just thought you might want to
know every 12 minutes, the hour hand moves one minute mark.
An example of this is 1:14.  When it's 1:14 and about 40 sec., you
can see that there is a little bit more than 7 minute marks
between the minute and hour hand (maybe 7 1/2).  This means that
when its 1:14 and about 40 sec., the 2 hands form a 45 degree
angle.

3. The angle that will be formed at 3:20 by the hands of a clock is a 20 degree angle.

Every time the clock passes 120 slashes the minute hand
moves 2 slashes down(2 minutes).Every time the minute hand moves
2 minutes the hour hand will move 1 degree.

Since there are 60 minutes in an hour / the amt. of minutes it
takes to move the hour hand one degree will give you the amt. of
degrees the hour hand will move in one hour. It is 30 and
30/ 6(degrees in one slash) = 5.  5 is the amt. of slashes between
each hour number.

From: ranierm@firnvx.firn.edu
Joshua Bellotti, Austin Clark, Molly Wilson, Oth Khotsimeuang
Bay Point Middle School, St. Petersburg, FL

Concerning problem number 1, we found that there is 6 degrees
between each minute marker because there is 60 minute markers on
the clock and 360 degrees.  So 360/60=6 degrees between each
minute marker.  Using a clock we found the following times that
form right angles:

12:16:08
12:48:50
1:21:32
1:54:14
2:26:56
3:00:00
3:32:42
4:05:24
4:38:06
5:10:48
5:43:30
6:16:12
6:49:00
7:21:42
7:54:24
8:27:06
9:00:00
9:32:42
10:05:24
10:38:06
11:10:48
11:43:30

There are 22 times every 12 hours and 44 times a day that the
hands of a clock form a 90 degree angle.  We found a pattern after
the first few times. There are 32 min, 42 sec, between each time.

Concerning problem number 2, we found that at 3:08 the hands of a
clock form a 45 degree angle.  We found this because we knew that
there are 60 minute markers all the way around a clock and 360
degrees around a clock.  So 360/60 = 6 degrees between each minute
marker.So you divide the number of degrees by 6, and that is how
many minute markers there are between the hands to form that
angle.  45/6 = 7.5, so there needs to be 7.5 minute markers
between the hands to form a 45 degree angle.  We looked at a clock
and found that at 3:08 there is 7.5 minute markers between the
hands.

Concerning problem number three, at 3:20 the hands form a
21-degree angle.  This is because there is 3.5 minute markers
between the hands at that time.  So 3.5 X 6 = 21 degrees.

From: ranierm@firnvx.firn.edu
Jim Montante, Kori Rowe, Clarissa Smith
Bay Point Middle School, St. Petersburg, FL

1. There are 22 different times that a right angle will occur in a
12 hour period. They are:

12:16:22, 12:49:05, 1:21:49, 1:54:33, 2:27:17, 3:00:00, 3:32:44,
4:05:28, 4:38:12, 5:10:55, 5:43:39, 6:16:23, 6:49:07, 7:21:50,
7:54:34, 8:27:18, 9:00:00, 9:32:45, 10:05:40, 10:38:13, 11:11:07,
and 11:43:40.

angle=>90/360=11/12t, 1/4=11/12t, 12/11(1/4)=11/12(12/11),
12/44=t, 3/11=t, 3/11(60)=180/11=16.3636min =>.3636(60)=21.82 sec

270/360=11/12t, 12/11(3/4)=t, 36/44=t, 9/11=t, 9/11(60)=49.0909 min
.0909(60)=5.454 sec.

for every hour set of right angles you must add 5.46 minutes.
1 1/4=11/12t, 12/11(5/4)=11/12(12/11), 60/44=t, 1.36, or 1 hr.
21.8 min, 21.82-16.36=5.46 min.

2. A 45 degree angle is formed at 12:08:16. Assuming that the
clock is a perfect circle then there are approx. 7 1/2 minutes /
tick marks in between the hour and the minute hands.  On a clock
there are 60 minutes and a circle has 360 degrees in it.  If one
fourth of a circle is 90 degrees, and one fourth of an hour is
15 minutes, then 90 degrees occurs when there are 15 tick marks or
min. between the two hands.  Since 45 degrees is half of 90
degrees, and 90 degrees is 15 min., it follows that a 45 degree
angle occurs when the hands are approx. 7 1/2 min./tick marks
apart.

45/360=11/12(t)

3. 20o

3:20=3 1/3 o'clock=10/3
assume minute hand moves at a rate of 1 rph.
assume hour hand moves at a rate of 1/12 rph
1-1/12 = 11/12
x = 11/12t, = 11/12*10/3, x = 20.0o

From: ranierm@firnvx.firn.edu
Kuon Lo, Bryan Yabczanka
Bay Point Middle School, St. Petersburg, FL

(1) The hands of a clock will make a right angle 44 times in
24 hours.  The times are:

12:16:45, 12:49, 1:22, 1:54:30, 2:27:30, 3:00, 3:33:45, 4:05,
4:38:30, 5:11, 5:44, 6:16, 6:49, 7:22, 7:54:30, 8:27, 9:00,
9:33:45, 10:05, 10:38:30, 11:11, 1:44.

After this the times repeat because there are 24 hours in a day.

(2)  A time in which the hands of a clock make a 45 degree angle
is 3:08.  The method for finding a time at which the hands of a
clock form any given angle is:

X / 6 = Answer  (Where X is the given angle.)

A full circle has 360 degrees.  So if there are 60 spaces between
tick marks (representing the 60 minutes in an hour) on a clock,
360 / 6 will give you 6.  This means that each minute mark is
6 degrees of the circle, or clock.  So by dividing the given angle
(X) by 6, you get how many minute marks seperate the minute hand
with the hour hand.

(3) The angle that the clock will form at 3:20 is a 20 degree
angle.  The method for finding an angle formed by the hands of a
clock at any given time is:

First part: W (0.0833333) = X  (Where W is the number of minutes from the given time and X is the answer for the first part.)

Second part: X + Chart Number* = Y (The chart # is determined by
the hour number from the given time and Y is the answer for the second part.)

Third part: | Y - W | = Z  (Z is the answer for the third part.)

Fourth part: Z (6) = Final Answer  (In degrees.)

* CHART

12:00..................0
1:00..................5
2:00.................10
3:00.................15
4:00.................20
5:00.................25
6:00.................30
7:00.................35
8:00.................40
9:00.................45
10:00.................50
11:00.................55

We got the number 0.0833333 from 5 / 60, where 5 is how many
minutes there are in between two whole numbers on a clock and
60 is how many minutes are in an hour. The chart came from the
fact that when the minute hand is on a certain number on a clock,
so-and-so minutes have passed since the exact time that the hour
changed. For example, if the minute hand is on 8, then 40 minutes
have passed since the hour changed. In the fourth part, you
multiply Z by 6 because, as already mentioned, 6 is the number of
degrees in a full minute mark. In some cases where the number of
minutes is 0, like 3:00, X is the number that the chart above
gives for 3:00, which is 15. The chart number for the second part
is also 15. In some cases, the formula won't work with some times.
That's because on the clock, the two hands make an angle whose
measure is greater than 180. In that case the chart number for the
second part should be the hour number exactly opposite the given
hour number on a clock. For example, for 12:49, use 6:00 on the

From: ranierm@firnvx.firn.edu
Nicole Tillotson, Nicole Federico, Logan Biniak, Tom Poucher
Bay Point Middle School, St. Petersburg, FL

1. A clock will make a right angle 22 times in 12 hours. The hour
and on the clock moves one minute mark every 12 minutes, and there
must be 15 minutes between the hour and minute hands to make a
90-degree angle. A right angle will occur twice in every hour, and
they are ususally 33 to 32 minutes apart. There are two exceptions
to that, both of which happen when the right angles occur exactly
on the hour, at 3:00 and at 9:00.

The times at which the right angles occur are:

12:16:22, 12:49:05, 1:21:49, 1:54:32, 2:27:16, 3:00:00, 3:32:43,
4:05:27, 4:38:10, 5:10:54, 5:43:38, 6:16:21, 6:49:05, 7:21:49,
7:54:33, 8:27:16, 9:00:00, 9:32:43, 10:05:28, 10:38:10, 11:10:55,
11:43:38.

The same minute and second time occurs every six hours.

2. The hands of a clock make a 45 degree angle at 12:08:08.  To
find a time in which the hands of a clock form any given angle,
first take your given angle (let's say it's 36 degrees), and
figure out how many minute markers that angle has inside of it.
In a 36 degree angle, there are 6 minute markers because for each
minute, there is 6 degrees and 6 * 6 = 36. Then figure how many
degrees the hour hand would move in 6 minutes. The hour hand moves
half a degree every minute, so the hour hand would move 3 degrees
in 6 minutes. Then figure how long it would take the minute hand
to move 3 degrees, using this formula:

[(amount of degrees hour hand moves)/6] = x/60

Then plug in the numbers you know:   3/6 = x/60

Cross multiply 3 * 60 = 180 and 180 divided by 6 = 30.  It would
take the minute hand 30 seconds to move 3 degrees.  And you can
check that by knowing that it takes the minute hand 60 seconds to
move 6 degrees, so it would take it 30 seconds to move 3 degrees.
Then you take all the time you know already: 6 minutes for the
angle, and 30 seconds for the minute hand to move 3 degrees, and
add it to 12:00, and you get 12:06:30 and that is a 36 degree
angle.

3. When the hands of a clock hit 3:20, they form a 20 degree
angle.  The hour hand moves half a degree each minute, and there
are 6 degrees in every minute.  At 3:20, the hour hand has moved
10 degrees, putting it between minute 16 and 17, 4 degrees.  The
angle would then be 20 degrees.

From: ranierm@firnvx.firn.edu
Markella Balasis, Mike D'Amico, J.R. Norris
Bay Point Middle School, St. Petersburg, FL

1. The clock will form a right angle 44 times every 24 hours. We
know this because we found that a right angle is formed two times
every hour, except for 2 o'clock and 8 o'clock.  At those hours,
only one occurs because the next right angle would be formed on
the hour.

The right angles are:

3:00:00 9:00:00
3:32:42 9:32:42
4:05:24 10:05:24
4:38:06 10:38:06
5:10:48 11:10:48
5:43:30 11:43:30
6:16:12 12:16:12
6:48:54 12:48:54
7:21:36 1:21:36
7:54:18 1:54:18
8:27:00 2:27:00

And the same for the other 12 hour period.

Using our knowledge of the fact that there are 44 in a day, we
know there are 22 every twelve hours.  If you multiply 12 hours
times 60 minutes, you get 720 minutes in 12 hours.  Divide 720 by
22, and you get 32 minutes and 42 seconds, which is the
approximate time lapse between every right angle.

2. A 45 degree angle occurs at 4:30.  We found this out by looking
at the clock and noticing that there were 15 marks between every
right angle.  We also knew that 45 is half of 90.  Thus, leading
to the conclusion that for every 7 1/2 marks a 45 degree angle was
formed.

3. The clock forms a 20 degree angle at 3:20.  The formula for
finding the angle at a specific time is as follows:

x = total number of minutes (hours changed into minutes, plus the minutes)

y = number of hours (always a natural number)

6 = number of degrees the minute hand moves every minute

.5 = number of degrees the hour hand moves every minute

360 = total number of degrees in the clock

the angle = |(6x - .5x) - 360y|

From: ranierm@firnvx.firn.edu
Todd Maxwell, Adrienne Balboni, Ashley Smith, Curtis Olt
Period 2
School: Bay Point Middle School

1) The way that we figured this out is by a simply wonderful
formula.  Using this formula it was quite easy to get all of the
right angles throughout the course of a single day.  This formula
is not a necessity to solve this problem, but it does help out
quite a bit.

This is the formula:

A)    x = 11 t   For the purpose of this formula x = 90
360      12                                  t = time

B)   90 = 11 t   90 degrees is the amount of that we're trying to
find out
360      12  how much time to add to each hour for a R.A.

C)    1 = 11 t   90/360 is the same as 1/4
4      12

D) 12 * 1 = 11 * 12 t   Next you multiply each side by the reciprocal
11   4   12      11  of 11/12 to get rid of 11/12.

E)  3 = t        12/11 * 1/4 = 3/11
11

F)  3 * 60 = t   Then, multiply 3/11 by 60/1 because thereare 60
11   1   minutes in an hour.

G)  180 = t      60/1 * 3/11 = 180/11
11

H)  16.3636      In a decimal form that gives you 16.3636.

I)  3636 * 60    You drop the 16 because 16 is the amount of min.
to 12:00 as the starting time.
i.e.. 12:00 + 16 = 12:16.__

J)  .3636 * 60 = 21.816  21.816 is the amount of sec. So, 12:16.22

K)  After you get your first right angle( 12:16.22 ) from then on
each hour after that one (i.e. 1:16.22) you add 5 minutes and 27
or 28 (depending on the circumstances) seconds.  After you add
5 minutes and 27 seconds to that hour you get 1:21.49 and after
that you get 2:21.49 + 5.27 = 2:27.32 and so on.

L)  Using this method you get all of the odd numbered right angles
such as 1,3,5 etc. (located on the right angle chart)

M)  To get the even numbered R.A.'s start with 12:__.__  Then plug
in 270 into the formula to get the starting the min. and sec.
which is 49.05 and then you add 5 minutes and 27or 28 (depending
on the circumstances) seconds. (i.e. 12:49.05; 1:54.32)

The other method to find out the right angles is by doing it
manually as well as visually.  This is how you do that. If you
were to try to find the first right angle in 12:00 doing it
manually and visually first you need to know that the hour hand
moves a 1/2 degree each minute and .0836 degree every 10 seconds
and the minute hand moves 6  degrees each minute, 1 degree every
10 seconds and .1 degree every second.  Next you need to know that
12:15 would be a right angle if the hour hand didn't move but
since it does you need to figure out how many degrees the hour
hand moved in 15 minutes.  Then you have to go back to the minute
hand and figure how long it would take the minute hand to move 7.5
degrees because that's how many degrees the hour hand moved in 15
minutes.  After you find out that it takes the minute hand 1 min.
and 15 seconds to move 7.5  degrees.  Next you must figure out how
many more degrees the hour hand moved in 1 min. 15 seconds which
is approximately .6 degree.  Then go back to the minute hand and
figure out how long it takes the minute hand to move .6 degrees.
It takes the minute hand 6 seconds to move .6 degrees.   After you
figure that out it breaks the amount of time down too far for the
hour hand to move at least .1 degrees.  So your final answer for
the first right angle in 12:00 is 12:16:22.

We would like to thank Mr. Montante, Jimmy Montante's father, for
helping us discover the formula that we used.

This is the right angle chart.

1.  12:16.22
2.  12:49.05
3.  1:21.49
4.  1:54.32
5.  2:27.32
6.  3:00
7.  3:32.43
8.  4:05.27
9.  4:38.10
10.  5:10.54
11.  5:43.38
12.  6:16.21
13.  6:49.05
14.  7:21.48
15.  7:54.32
16.  8:27.15
17.  9:00
18.  9:32.43
19.  10:05.28
20.  10:38.11
21.  11:10.55
22.  11:43.38

After you find these times of R.A.'s you have to double it because
there are 24 hours in a single day, not 12.  After doing that you
find that there are 44 right angles throughout the day.

2) a. An example of a 45 degree is 12:08.10
b. The method to find this is by plugging 45 into the formula
from problem 1 and by following the all of the steps.

3) a. 20 degrees
b. The way that you can find this angle at any given time is by
knowing that the hour hand moves 1/2 degree every minute.  At 3:20
if the hour hand didn't move then it would be a 30 degree angle;
however, the hour hand does move 1/2 degree each minute.  So,
after 20 minutes have gone by the hour hand has moved 10
degree(because 20 * 1/2 degree = 10 degrees) which decreases the
size of the angle to 20 degrees instead of what would be 30
degrees.

From: jyoung@cello.gina.calstate.edu>
Marty Klein, Jacqulyne Law, Ki Chang
College Park High School, Pleasant Hill, California

1. There are 22 right angles that the clock will make during the
course of a day.  For the record, there are:

12:16, 12:48, 1:22, 1:54, 2:28, 3:00, 3:32, 4:06, 4:38, 5:12,
5:44, 6:18, 6:50, 7:22, 7:56, 8:28, 9:00, 9:34, 10:06, 10:38,
11:12, and 11:44.

We found these times by turning the knob found on the back of an
analog alarm clock. Since the hour hand moves along with the
minute hand, we found this to  be the most accurate method of
finding the times we needed.  If we were to use  any other method,
we may not have accounted for the hour hand moving along with  the
minute hand (for instance, 6:15 is NOT a right angle, as some
people will  say).

Now for a pattern.  First we looked at our first few times, 12:16,
12:48, and  1:22.  48-16=32, so the first two times are 32 minutes
apart.  The difference  between the next two times (1:22 and
12:48) is 34.  Ok, we noted that these two  numbers are very
similar.  Lets see if a pattern continues.  Our next time,  1:54
is 32 minutes later than 1:22.  Ah ha!  Maybe they alternate
between being  32 and 34 minutes apart!  We quickly checked the
next time by simple adding 34  minutes to 1:54.  Yes, it IS 2:28!

At this point, we were pretty confident  that we had discovered
this elusive pattern.  Not so fast. When we applied our  little
pattern to all of the other times, it seemed to work, but, alas,
there were some exceptions. First, 3:00 and 3:34 are 34 minutes
apart, but according  to our pattern, they were supposed to only
have a 32 minute difference. D'oh!

Another flaw occurred between the times 6:50 and 7:22. Where the
difference is still only 32 minutes, it was its turn to be 34
minutes apart. All told, there are four exceptions to our rule,
but all of them are either 32 or 34 minutes apart. We thought we
had the clock beaten when we discovered this whole alternation
theory, but I guess we spoke too soon. We got close, anyway.

2. We were able to determine that 4:00 would be pretty close to a
45 degree angle, although we had no luck in generalizing our
method.

3. At 3:20, the hands of the clock will be at an approximate 20 degree
angle. Here is how we determined this:

We figured that the whole clock has 360 degrees total (one circle).
Generally, 3:20 would span from the three (little hand) to the four
(big hand), or 1/12 the entire clock. Follow that? This angle
measure works out to be 30  degrees. As we have mentioned before,
we must remember that the hour hand moves [ever so slowly], also.
this must be accounted for. Since :20 is 1/3 an  hour, we figured
that it moved exactly 1/3 the way to the four, which would form a
20 degree angle. You see?

Again, generalization is not in our composite vocabulary, so we were
unable to do the second part.

From: anne-d.-sandler@shhs1.ccsd.k12.co.us
From: John MacArthur
School: Smoky Hill High School

The answer to number 3 is 26 degrees.  Each minute mark is 6 degrees,
so between the three and the four there are 36 degrees.  To find out
how far the hour hand moved past the three, you divide 20 minutes by
2.  That means that you subtract 10 degrees from 36.

From: anne-d.-sandler@shhs1.ccsd.k12.co.us
Amber and Meagan
School: Smoky Hill High School

2. The angle of the hands of the clock at the time of 3:20 is 20
degrees or an acute angle.

3. The time that the hands of the clock form a 45 degree angle is
4:30.
To find the degrees of an angle at any given time would be to know
that in between each hour is 30 degrees and each minute within the
hour is 6 degrees.

You also have to know that as the minute hand moves so does the hour hand move.

From: anne-d.-sandler@shhs1.ccsd.k12.co.us
Somsnit Vanprapa and Kary Wyell
School: Smoky Hill High School

times the hands of the clock will form 90 degrees is 22 different
times.

The second question is at what times they will occur. The following is the list of times.

12:16:21; 12:49:05; 1:21:49; 1:54:32; 2:27:16; 3:00; 3:32:43;
4:05:27; 4:38:11; 5:10:55; 5:43:38; 6:16:21; 6:49:05; 7:21:49;
7:54:32; 8:27:16; 9:00; 9:32:43; 10:05:27; 10:38:11; 11:11:55;
11:43:38.

Then you asked to find the time at whick the hands of a clock make
a 45-degrees angle. The time is 4:30.  To do this problem, I used
a very simple method. That is to use a real clock. However, we
would better keep something in mind in order to make sure that our
answer is right.  From the basic idea that a section of a clock is
30 degrees, I thought that ot get a 45 degrees angle then I would
have to add another 15 degrees.  The problem is that the hour hand
of a clock moves while the minute hand moves; therefore, we cannot
easily add the 15 degrees right away.  The real clock experiment
would really help a lot to make it easilier.  I moved the  hands
of the clock around it by keeping the idea that a section of a
clock is 30 degrees and half of it is 15 degrees.  I came up with
4:30 as a result.

The hands of the clock will form a 20 degrees angle  when it is 3:20.
To do this problem, it will be very helpful to draw a picture or a
diagram to illustrate it.  Draw the hands of the clock at the time
given. It is important to know that each space or section between
each number is 30 degrees. After finishing drawing the hands of the
clock, count how  many sections (from number to mumber) the interior
parts made by the hands of the clock at the time given. If the time
given is not exact, you may count from the original time (like this
problem of 3:20, start form 3' clock first.) Then use the logic to
finds how many degrees the hour hand has moved within one section
from the original time (like this problem, we know that it is 20
minutes past 3 o'clock: we shall then find out how many degrees there
are in  20 minutes.) This is stated in a formular H/60 * 30 or H/2
(H stands for the minutes the hour hand has moved from the orginal.

Now you will subtract the degrees formed by the hour hand and the
original time form your total degrees. For instance, like this
problem, our total degrees would be 30 degrees because we started
from the orginal time whick is 3' clock and ended at number 4
because it was 20 minutes passed 3 o'clock. We know that the hour
hand is not exactly at number 3 because it is 20 after. To find how
many degrees from  orginal 3 o'clock to 20 minutes after, we used
formular H/2; 20/2 = 10; then we know that it is 10 degrees from 3
o'clock to the spot where it has moved to 20 minutes after. To find
how many degrees the hands of the clock formed, we subtract the
degrees we don't want from the total 30-10 = 20 degrees.

From: anne_d._sandler@sshs1.ccsd.k12.co.us
From: Jennifer Sprangers, Yunny Chen
School: Smoky Hill High School

1. For the first part of this question: At how many different
times will the hands of a clock make a right angle?

The answer is 22 times. The times that this will occur are:

12:16:21, 12:49:05, 1:21:49, 1:54:32, 2:27:16, 3:00:00, 3:32:43,
4:05:27, 4:38:11, 5:10:54, 5:43:38, 6:16:21, 6:49:05, 7:21:49,
7:54:32, 8:27:16, 9:00:00, 9:32:43, 10:05:27, 10:38:11, 11:10:54,
11:43:38!

2. A time where the hands of a clock form a 45 degree angle is
4:30.

3. The angle formed by 3:20 is 20 degrees.  This is how we arrived
at our answer (m= minutes past 12:00, going clockwise):  We set up
an equation that looked like this: measure of angle = 6m + or -
m/2 - number of degrees to the first clock hand from 12 'clock
going clockwise, rounded to the earliest hour.  The reason we have
a + or - m/2 is because if you meet the hour hand first, you
subtract m/2.  If you meet the minute hand first, you add m/2.
For example, to find the angle formed by 3:20, we drew a clock.
Then plugged in the time with the clock hands, hour hand at three,
and minute hand at twenty minutes.  6m (6 times 20) is 120.  We
subtracted m/2 (20/2) because the hour hand came first.  6m - m/2
= 110.  Then we need to subtract the number of degrees (from 12
0'clock) it takes to reach the first clock hand, rounding it to
the earliest hour, which is 3.  6m - m/2 - 90 = 20.  And that is
how we arrived at twenty degrees.

From: dmccarty@mail.wiscnet.net
From: Meagan Cihlar
School: Sturgeon Bay High School

1 - 9:00pm, 9:00am, 3:00pm, 3:00am, 110:37am, 10:37pm.

2 - 3:07.30  Between any two consectutive numbers on a clock
face including 12-1 consectutive these are 30 degrees.

3 - 30 degrees.  Between each number (say 2 and 3) there is 30
degrees.

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