Pentagon Area Using No Trig

Date: 05/14/2001 at 12:49:05
From: Sandra
Subject: Pentagon Area - Using NO Trig

Dr. Math,

I've been trying to find the area of a pentagon without the use of 
trigonometry. Where I am stumped is in finding the area of one of the 
five triangles. I can use trig to find the height of the triangle very 
easily, but the challenge is finding the height and thus the area 
without the use of trig. I also know the area of a pentagon is 
1.720x^2, but I don't know the derivation of this equation either. 
Please help.

Thank you,
Sandra

Date: 05/14/2001 at 17:04:04
From: Doctor Rick
Subject: Re: Pentagon Area - Using NO Trig

Hi, Sandra, and thanks for writing to Ask Dr. Math.

Perhaps you've seen the item in our Dr. Math Archives about Finding 
the Area of a Regular Pentagon using trigonometry:

   http://mathforum.org/library/drmath/view/54071.html   

Trigonometry isn't necessary, however. We can find the area without 
it.

First draw the diagonals of the pentagon. Each of the five diagonals 
is divided into three segments by the other diagonals. The outer two 
segments are equal in length; call their length a. Call the length of 
the center segment b. The length of a side of the pentagon is s.

You can prove that certain triangles formed by the pentagon and its 
diagonals are similar. For one, the triangle formed by two adjacent 
sides and one diagonal of the pentagon is similar to the smaller 
triangle formed by one of these sides and two segments of length a.

Also, you can prove that the triangle formed by one side, one segment 
of length a, and one segment of length a+b is isosceles. Therefore, 
a + b = s.

Using these facts you can prove that the length of the diagonal of the 
pentagon is (sqrt(5)+1)/2. I will call this length d.

Now we're ready to find the area of the pentagon without using 
trigonometry. Draw the two diagonals from the top vertex of the 
pentagon. These divide the pentagon into three isosceles triangles. 

You can find the altitude of each of these triangles using the 
Pythagorean theorem and the fact that the altitude of an isosceles 
triangle bisects the base. Now you know the altitude and base of each 
triangle, so you can find the area as half the product of the base and 
the altitude. 

Alternatively, you can use Heron's formula for the area of a triangle 
in terms of the side lengths alone. I used this approach.

The sum of these areas is the area of the pentagon. I don't get a 
really neat formula, however. I get this formula using the value of d, 
which I'm leaving for you to calculate:

     K = (sqrt(4d^2 - 1)/4 + d*sqrt(4 - d^2)/2)s^2
       = 1.720477 s^2

Perhaps you could calculate d, plug the value of d into the formula 
for K, and find a way to simplify the formula. I'll keep playing with 
it myself.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/15/2001 at 18:36:33
From: Sandra Ward
Subject: Re: Pentagon Area - Using NO Trig

Dr. Math,

I wanted to thank you for your timely response and insightful 
assistance to the pentagon problem. It certainly helped me out a 
bunch!

Sandra