Formula for Area of Any Regular Polygon

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/