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### Formula for Area of Any Regular Polygon

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Date: 03/01/98 at 20:42:54
From: Andrew Miklas
Subject: 3D Geometry

Is there a formula to calculate the area of a regular polygon that
uses the number of sides and the length of a side as the inputs?

For example,

Sides      Length of Side     Area
____________________________________

3                 6            18
4                 8            64
5                 2             ?
10                13            ?
100               25            ?

If so, it would follow that there is a formula that calculates the
volume and surface area of such a figure (when it is put into prism
form), where

x = formula
l = prism length
s = side length
n = number of sides

x * l = volume
s * l * n + 2x = surface area

Am I correct?
```

```
Date: 03/02/98 at 17:04:27
From: Doctor Sam
Subject: Re: 3D Geometry

Andrew,

You are correct that such a formula exists, and that the total surface
area of a regular prism is given by your formula. But the numbers in
your table are not all correct. The area of an equilateral triangle
of side 6, for example, is 9 * sqrt(3), not 18.

The formula that you are looking for is derived by taking your regular
polygon and dividing it up into triangles in a special way.

A regular polygon can be inscribed in a circle . . . just imagine the
center of the polygon as the critical point P. Connect P to each of
the n vertices of the polygon . . . this divides it into n isosceles
triangles.

The height of each triangle needs to be computed using trigonometry.
Since there are n triangles, the vertex angle of each triangle (at P)
is 360/n degrees. The altitude of the triangle bisects the angle,
giving an acute angle of 180/n degrees.

This lets us compute the altitude h as the tangent of 180/n. The side
opposite the angle is half the base (half the side of the polygon) and
the adjacent side is the unknown height. This gives:

tan(180/n) = s/(2h)
or
h = s/(2 tan(180/n))

The area of this isosceles triangle is, therefore,

A = (1/2) * s * h
=  s^2/(4 tan(180/n))

Since there are n triangles, the area of the n-sided regular polygon
is

x(s) = n s^2/(4 tan(180/n))

Notice that if n=4, this reduces to the area of a square, since
tan(45) = 1; and if n=3, this reduces to the area of an equilateral
triangle, since tan(60) = sqrt(3).

I hope that helps.

-Doctor Sam, The Math Forum
Check out our web site http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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