Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search