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Finding the Area of a Regular Pentagon


Date: 04/15/98 at 20:06:08
From: Jon Gilman
Subject: The area of a regular pentagon

How can you find the area of a regular pentagon only knowing the 
length of one side? Is there a way to do it without using 
trigonometry? If so, how? Please help me with this question.

Thanks.
Jon


Date: 06/17/98 at 12:33:27
From: Doctor Naomi
Subject: Re: The area of a regular pentagon

Hi Jon,

I don't know of a way to easily get the area of a regular pentagon 
without using trigonometry.

What I'd suggest is that you draw out your pentagon and divide it up 
into 5 identical triangles where each side of the pentagon is the base 
of a triangle (all of the triangles share the center as a vertex).  
Then all we have to do is find the area of one of these triangles and 
multiply by 5 to get the area of the pentagon.

How do we get the area of one of these five triangles?  Let's let the 
length of each pentagon side (and thus each triangle base) equal 2x.  
Our triangle looks like this (I've given it vertices A, B and C where 
A is at the center of the pentagon):

          A
          /\
         /  \
        /    \
       /      \
      /        \
     /__________\
   B      2x      C

To find the area of this triangle, we are going to need to know the 
height (we already know the base = 2x). Now what? It doesn't seem 
clear how we can find the height. But don't forget that this triangle 
is 1/5th of a regular pentagon, which this means that 5 angle A's add 
up to equal 360 degrees. Can you see why?

So angle A = 360/5 = 72 degrees. How can we get angles B and C?  
Because our pentagon is regular, side AB = side AC, which means
angle B = angle C. Let's let angle B = angle C = n degrees. What is n? 
Recall that the sum of the angles in a triangle equals 180 degrees.

To get n, you'd have to solve: 

   180 = sum of angles A, B and C
       = 72 + n + n

To get the height we just need to use a little trig. Let's let the 
height equal h. By drawing in h, we cut angle A in half. I've added a 
new point D in the diagram below:


           A
          /|\
         / | \
        /  |  \
       /  h|   \
      /    |    \
     /_____|_____\
   B    x  D   x  C

Now we know the measure of angle DAC (we cut angle A in half), of 
angle C (equals n from above) and angle ADC (the height is always 
perpendicular to the base).  

Finally, it's time for some trig. You can calculate the tangent of 
angle C using a calculator (remember to use degrees) and then solve 
for h in terms of x using the following:

   tangent C = opposite side/adjacent side = h/x.

Once you have h in terms of x, you can calculate the area of triangle 
ABC, and 5 times this area is the area of the pentagon.

I hope this helps you out!  

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

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