Forming a Regular Pentagon by Folding Paper

Date: 3/11/96 at 19:5:8
From: Anonymous
Subject: Help!

I need help on this brain teaser.  Is it possible to form a 
regular pentagon from a strip of paper of any size by just 
folding?  If so how and where do you fold the paper.  Does the 
paper need to be a certain size?

Date: 3/21/96 at 13:5:41
From: Doctor Aaron
Subject: Re: HELP!!!

Here are some hints that I hope will help you.

Draw the pentagon on a piece of paper.  Now draw a rectangle 
around the pentagon so one of the sides of the pentagon is sitting 
on one of the sides of the rectangle, and each of the corners (or 
vertices) of the pentagon touches a side of the rectangle.  You 
should see a pentagon and 4 triangles that make a rectangle 
together.

From this we can see that all we have to do is fold the strip to 
the proper dimension rectangle, and then fold a bunch of corners 
inward.

We still have to see exactly how to fold the corners, and what the 
dimensions of our rectangle should be.

We know that the sides of the regular pentagon are all the same 
length, call it a.  We also know that the interior angles are 
equal to each other and are 108 degrees.  We get this value by the 
relation:

(total angle in a polygon in degrees)) = 180*(number of sides - 2).

If this does not make sense to you, consult a geometry text or 
teacher, or write back and we'll explain.

Now we have a bunch of angles.  Trigonometric relationships are 
going to give us the sides of the triangles on the outside of the 
pentagon, which will get us the sides of the rectangle.

Use the interior angles of the pentagon to get the exterior angles 
of the pentagon, which are also the interior angles of the 
triangles. Start with the side of the pentagon that overlaps the 
side of the rectangle, and work your way around the pentagon until 
you know what all of the angles are.  Remember that the sum of the 
angles whose composition is a line is 180 degrees.  

Once you have all of the angles, note that the hypotenuse of each 
triangle is a side of the pentagon, and use trigonometry to get 
the sides of the rectangle.

You now have the size of the rectangle in terms of the side of the 
pentagon, and the lines along which you have to fold.

Good luck!

-Doctor Aaron,  The Math Forum