Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » Education » math-learn

Topic: [math-learn] Stem & Leaf Plots with Decimals
Replies: 21   Last Post: May 5, 2008 5:31 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Ladnor Geissinger

Posts: 313
From: University of North Carolina at Chapel Hill
Registered: 12/4/04
[math-learn] length and area; sidewalks and roads
Posted: May 5, 2008 12:45 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply
att1.html (14.9 K)

In response to several recent email notes about computing areas of
simple geometric regions, I will resend my note to math-learn from Nov
2003. That original note was in response to comments by Rex Boggs. I
hope he forgives me for throwing it out there again.
================================================================
Rex Boggs complained about his students getting mixed up about
circumference and area and remembering formulas and knowing which to
use when. Perhaps some teachers would find it useful to develop the
idea of computing area from two common examples of geometric regions:
constant width sidewalks (and borders and ribbon paths) and roads.
For each of these the computation of area is simple: multiply the
constant width w by the centerline length m.

Note that for any trapezoid with width w (distance between
parallel sides) and mid-line length m, the area is w*m. And borders
around parts of buildings, patios, and other objects are often made up
of a collection of such trapezoids all of the same width w and with
two successive trapezoids meeting at a corner along a line which makes
the same angle with each edge at the corner. Sidewalks are usually
made this way. So the total area of such a sidewalk or border is the
product of the constant width w multiplied by the sum of the mid-line
lengths of each of its segments.
Also, suppose we take a long piece of ribbon and tape a string to its
center-line and then start pasting in down flat onto a
surface. When we want to have it go off in a different direction we
cut the ribbon but not the string and flip the ribbon over and
continue with the two cut edges realigned. Clearly the total area of
our ribbon path is the width multiplied by the total mid-line length
of the string (no matter at what angle the ends are cut). Of course a
special case is a single triangle with w the "altitude" from vertex C
dropped to side AB (possibly extended), and the area is w*m where m is
the length of the midline parallel to AB and halfway between AB and C.
So our "formula" works for rectangles, squares, triangles,
trapezoids,..-- sidewalks and pieces of them.

Roads of constant width w are nearly a series of rectangles of width w
connected by sectors of circular rings. In this case again, the total
surface area of such a road is the product of w and the center-line
length. Does this kind of area computation appear in any texts? Note
that special cases include a sector of a ring or the whole ring for a
circular race track, and the really special case of a sector of a
circle or a whole circle. Note that one could argue that this works
for roads by approximating closely a road by using a ribbon path.
So we don't have to know about pi*rad^2 before doing this.

On the other hand, if we do know a formula for the area of a circle,
we could consider two concentric circles with radii a>b. Then for any
sector with angular measure t, the area of the part of the ring
between the two circles and within the sector is
(t/2pi)*pi*a^2 - (t/2pi)*pi*b^2 or
(a-b)*t*(a+b)/2. And a-b is the constant width of that ring while
t*(a+b)/2 is the length of the midline, that is, the arc of the circle
of radius (a+b)/2 which lies in this sector. So our simple formula for
area has appeared again.




Date Subject Author
4/9/08
Read [math-learn] Stem & Leaf Plots with Decimals
Jon Wilson
4/10/08
Read [math-learn] Re: Stem & Leaf Plots with Decimals - FOUND!
Jon Wilson
4/10/08
Read RE: [math-learn] Re: Stem & Leaf Plots with Decimals - FOUND!
Prof Martin Weissman
4/10/08
Read RE: [math-learn] Re: Stem & Leaf Plots with Decimals - FOUND!
Ron Ferguson
4/10/08
Read [math-learn] Re: Stem & Leaf Plots with Decimals
Prof Martin Weissman
5/4/08
Read Re: [math-learn] Re: Stem & Leaf Plots with Decimals
lolly45101961
5/4/08
Read Re: [math-learn] Re: Stem & Leaf Plots with Decimals
Ed Wall
5/4/08
Read RE: [math-learn] Re: Stem & Leaf Plots with Decimals
Marie Bahlert
5/4/08
Read [math-learn] Area of a trapezoid
Rex Boggs
5/4/08
Read Re: [math-learn] Area of a trapezoid
Ed Wall
5/5/08
Read [math-learn] length and area; sidewalks and roads
Ladnor Geissinger
5/5/08
Read Re: [math-learn] length and area; sidewalks and roads
GS Chandy
5/5/08
Read Re: [SPAM]Re: [math-learn] length and area; sidewalks and roads
Rex Boggs
5/5/08
Read Re: [math-learn] length and area; sidewalks and roads
Ralph A. Raimi
5/5/08
Read [math-learn] Re: length and area; sidewalks and roads
Ladnor Geissinger
5/5/08
Read Re: [math-learn] Re: length and area; sidewalks and roads
Ralph A. Raimi
5/4/08
Read Re: [math-learn] Re: Stem & Leaf Plots with Decimals
Ralph A. Raimi
4/10/08
Read RE: [math-learn] Re: Stem & Leaf Plots with Decimals - FOUND!
Mr Anderson
4/10/08
Read [math-learn] Re: 5 point summary, IQR
Prof Martin Weissman
4/10/08
Read Re: [SPAM][math-learn] Re: 5 point summary, IQR
Rex Boggs
4/12/08
Read RE: [SPAM][math-learn] Re: 5 point summary, IQR
Marie Bahlert
4/10/08
Read [math-learn] Re: Stem & Leaf Plots with Decimals - FOUND!
Jon Wilson

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.