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### Building a Circular Horse Pen

```Date: 06/16/2002 at 22:05:37
From: Larry Roberts
Subject: Question about building a circular horse pen

Dr. Math,

My Dad and I are building a round pen for our horse.  We have 16
16ft. panels and a 10 ft. gate and a 4ft. gate (270 ft. total).  We
want to use a radius and mark the places to dig holes for each post
that will support the panels, but we don't know how long the radius

Larry Roberts
```

```
Date: 06/17/2002 at 10:44:07
From: Doctor Rick
Subject: Re: Question about building a circular horse pen

Hi, Larry. This is an interesting practical geometry/trigonometry
problem!

We can get a quick approximation by pretending that the lengths you
name are actually arc lengths around the circumference of the circle.
The circumference is then 270 feet, as you say. The circumference of
a circle is 2 pi (about 6.28) times the radius of the circle.
Therefore the radius is the circumference divided by 6.28:

C = 2*pi*R

C/(2*pi) = R

Dividing 270 feet by 6.28, we get very close to 43 feet.

Now let's do it in a more accurate way. The lengths are really chord
lengths, or sides of a polygon inscribed in the circle. If the radius
of the circle is R, then we can imagine drawing an isosceles triangle
with the two equal sides of length R and the third side (the base)
equal to the length L of the chord (16 feet for each panel, etc.)
Using trigonometry, the apex angle (the angle opposite the base) is

theta = 2*arcsin(L/(2R))

The entire circle can be pictured as 18 isosceles triangles with
their apexes together at the center. The sum of the apex angles must
be 360 degrees.

360 = 32*arcsin(8/R) + arcsin(5/R) + arcsin(2/R)

Solving this equation for R is not easy! I'm just going to do it by
successive guesses, since we have a pretty good first guess, R=43.
Try that number to see what angle we get:

32*arcsin(8/43) + arcsin(5/43) + arcsin(2/43) =
343.11 + 6.67 + 2.67 = 352.45 degrees

That's 7.55 degrees too small. We want to increase the angle by a
factor of 360/352.45 = 1.0214. Let's try decreasing R by this ratio
(so each angle will be bigger):

R = 43*(352.45/360) = 42.098 degrees

32*arcsin(8/42.098) + arcsin(5/42.098) + arcsin(2/42.098) =
350.550 + 6.821 + 0.048 = 357.419 degrees

That's closer. When I put the calculations in a spreadsheet so I can
repeat this process as often as I need, I find that after only 4
calculations I get very close to 360 degrees, with a radius of
42.10892 feet. That's much more accurate than you need!
To convert to feet and inches, we just multiply the fractional part
by 12 inches per foot:

0.10892 foot * 12 inches/foot = 1.30 inches

The radius is therefore 43 feet, 1.3 inches.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/17/2002 at 22:23:27
From: Larry Roberts
Subject: Thank you (Question about building a circular horse pen)

Dr. Math,
Thank you so much.  My Dad was told that 43 ft. would be
the length of the radius, but he wanted me to make sure.
It's been a long time since I've done problems like this
and didn't want to disappoint him (on father's day) by
digging all these holes to find out we were wrong.  Your
help is greatly appreciated.  You're the best!
Larry Roberts
```

```
Date: 06/18/2002 at 08:10:04
From: Doctor Rick
Subject: Re: Thank you (Question about building a circular horse pen)

Hi, Larry.

Uh-oh, I just looked back at my answer and noticed a major typo in
the last line -- the answer should be 42 feet, 1.3 inches, not 43
feet, 1.3 inches! I hope you haven't started digging, or that you
noticed my blunder!

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/23/2002 at 19:03:34
From: Larry Roberts
Subject: Question about building a circular horse pen

Are you serious?  It's 42 ft not 43 ft?  We've already set some posts
at 43 ft.  Is there any way that we could do a portion at 43 ft. and
then decrease a portion of the circle to another length to make the
circle work?
```

```
Date: 06/24/2002 at 10:01:31
From: Doctor Rick
Subject: Re: Question about building a circular horse pen

Hi, Larry.

I'm sorry. Not only did I make that typo (saying 42.10892 feet = 43
feet 1.3 inches), but I had an error in the calculation early on -- I
failed to multiply two of the arcsines by 2, as I did for the
calculation of the angle subtended by each 16-foot panel. But there's
some good news at the end. Here is the entire explanation again,
corrected:

*****

We can get a quick approximation by pretending that the lengths you
name are actually arc lengths around the circumference of the circle.
The circumference is then 270 feet, as you say. The circumference of
a circle is 2 pi (about 6.28) times the radius of the circle.
Therefore the radius is the circumference divided by 6.28:

C = 2*pi*R

C/(2*pi) = R

Dividing 270 feet by 6.28, we get very close to 43 feet.

Now let's do it in a more accurate way. The lengths are really chord
lengths, or sides of a polygon inscribed in the circle. If the radius
of the circle is R, then we can imagine drawing an isosceles triangle
with the two equal sides of length R and the third side (the base)
equal to the length L of the chord (16 feet for each panel, etc.)
Using trigonometry, the apex angle (the angle opposite the base) is

theta = 2*arcsin(L/(2R))

The entire circle can be pictured as 18 isosceles triangles with
their apexes together at the center. The sum of the apex angles must
be 360 degrees.

360 = 32*arcsin(8/R) + 2*arcsin(5/R) + 2*arcsin(2/R)

Solving this equation for R is not easy! I'm just going to do it by
successive guesses, since we have a pretty good first guess, R=43.
Try that number to see what angle we get:

32*arcsin(8/43) + 2*arcsin(5/43) + 2*arcsin(2/43) =
343.11 + 13.35 + 5.33 = 361.79 degrees

That's 1.79 degrees too big. We want to decrease the angle by a
factor of 360/361.79 = 0.9950. Let's try increasing R by the inverse
of this ratio (so each angle will be bigger):

R = 43*(361.79/360) = 43.214 feet

32*arcsin(8/43.214) + 2*arcsin(5/43.214) + 2*arcsin(2/43.214) =
341.391 + 13.288 + 5.305 = 359.984 degrees

That's much closer. I can put the calculations in a spreadsheet and
repeat this process as often as I need; after 4 calculations I get
360.0000 degrees, with a radius of 43.21204 feet. That's much more
accurate than you need! To convert to feet and inches, we just
multiply the fractional part by 12 inches per foot:

0.21204 foot * 12 inches/foot = 2.544 inches

The radius you want is 43 feet, 2 1/2 inches.

*****

My errors canceled out. You are only off by 2 1/2 inches (1.2 inches
from the figure I gave you), not the 10.7 inches (in the other
direction) that my erroneous answer indicated! If you use a radius of
43 feet, the total angle is 1.79 degrees too high, resulting in an
overlap of

43*(1.79/180)*3.1416 = 1.34 feet

If you put half the posts at a radius of 43 feet already, then you
could put the other half at a radius of 43 feet, 5 inches, and you'll
come out about right. There is probably some imprecision in the
placement of the posts in the holes anyway, so may not be worth while
trying to be more accurate than this.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/25/2002 at 22:17:39
From: Larry Roberts
Subject: Question about building a circular horse pen

That sounds great.  Again thanks for your time.  We have already put
some posts in concrete and after I told my Dad we were off by a foot
he said he'd just buy an extra panel.  Now it looks like we can
continue as we were and we'll be ok.  Thank you.

Larry Roberts
```
Associated Topics:
College Conic Sections/Circles
College Geometry
College Trigonometry
High School Conic Sections/Circles
High School Geometry
High School Practical Geometry
High School Trigonometry

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