Expert: Paul Klarreich - 7/1/2007

Question
A regular pentagon is inscribed in a circle with a radius of 5 centimeters.  Find the length of a side of the pentagon.

Answer
Questioner:   sandra
Category:  Advanced Math

Question:  A regular pentagon is inscribed in a circle with a radius of 5 centimeters.  Find the length of a side of the pentagon.

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Hi, Sandra,

It's hard to do this without a diagram, so you will draw one from my instructions.  You will use specialized equipment that may be difficult to find in this computer age:  A PENCIL AND PAPER.

Draw this diagram:

a. Make your circle.  Mark the center C.

b. Draw your inscribed pentagon.

c. Select two consecutive vertices.  Call them A and B.  It works out best if they are at the bottom of the figure.

d. Draw AC and BC, making triangle ABC.

e. Note that angle ACB = 72d.  (72 degrees)

f. Draw the altitude from C to AB, meeting AB in D, the midpoint of AB.

Look at triangle ACD.  Observe the following about this triangle:

1. angle ACD is half of ACB, therefore 36d.
2. angle ADC is 90d, so this is a right triangle.
3. angle CAD is 54d.
4. side AC is the hypotenuse and is equal to 5.

Now AD = 5 cos 54d (which you compute with your calculator) and:
AC = 2 AD, and is the length of your side.