Expert: Paul Klarreich - 1/23/2007

Question
Hi, my question is that if you have a pentagon, which has five sides, and you fold it in half, witht the outside lenths of 1, what would be the length of the line going down the middle of the pentagon.

Answer
Questioner:   Manish
Category:  Advanced Math
 
Subject:  Advance algebra 2
Question:  Hi, my question is that if you have a pentagon, which has five sides, and you fold it in half, witht the outside lenths of 1, what would be the length of the line going down the middle of the pentagon.
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Hi, Manish,

Try the following diagram:
1. Construct your (regular) pentagon.  OK, don't construct it, just draw it.
2. Mark the center A, pick two consecutive vertices, and mark them B and C.
3. Find the center of BC and mark that D.
4. Draw AB, AC, AD.  AD is perpendicular to BC.

Now you should observe the following:

BC = 1, so BD = 1/2 = 0.5
Angle ABD = 54 degrees.
Angle BAD = 36 degrees.

Call AD = h, and AB = r.  
h + r is the length you are looking for.

Use some right triangle trigonometry:

h/BD = tan 54,  so  h = 0.5 tan 54
Once you have your value of h, use the pythagorean theorem to find r from  

0.5^2 + h^2  = r^2

You'll have to push a few buttons on your calculator, but that should do it.