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 
should be.  Can you help?  Please!  Thanks in advance.  


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