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
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.