Finding Areas of Different Polygons

Date: 09/02/97 at 16:49:08
From: alex
Subject: Geometry area working out on polygons

Could you please tell me how to work out the area for an equilateral 
heptagon, octagon, nonagon, decagon, unedecagon, and dodecagon? 

Thank you.

Date: 09/08/97 at 15:59:36
From: Doctor Rob
Subject: Re: Geometry area working out on polygons

For any regular n-gon, draw it and find the center of its 
circumscribed circle, which we will call O. Draw radii from O to two 
adjacent vertices of the n-gon, A and B. Then angle AOB will have 
measure 2*Pi/n.  

Drop a perpendicular from O to AB, bisecting it at C. Then the area 
of the entire n-gon will be n times the area of triangle OAB, or 
2*n times the area of triangle OAC. The base of triangle OAC is AC, 
whose length is s/2 (s is the side length of the n-gon). The altitude 
of triangle OAC is OC, whose length is (s/2)/tan(Pi/n), since angle 
AOC is half of angle AOB, or half of 2*Pi/n, or Pi/n. 

Thus the area of triangle OAC is (1/2)*(s/2)*(s/2)/tan(Pi/n), and the 
area of the n-gon is

  A = (n*s^2/4)*cot(Pi/n).

For n = 8, 10, and 12, the cotangent can be found in terms of 
radicals. For n = 7, 9, and 11, it cannot.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/