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Origami Equilateral Triangle

Date: 04/26/2001 at 00:06:02
From: Oliver
Subject: Folding Construction: Equilateral Triangle


I have spent quite a long time trying to figure out how to create an 
equilateral triangle by folding; not using the sides of a piece of 
paper and using nothing but your hands and, of course, your brain.

I have tried to make a hexagon from a piece of paper by folding, 
because that would give you six equilateral triangles, but I have had 
no luck in creating a hexagon from only folding.

I would appreciate your help and I am thankful you took the time to 
read this.


Date: 04/26/2001 at 12:36:34
From: Doctor Peterson
Subject: Re: Folding Construction: Equilateral Triangle

Hi, Oliver.

I'm not quite sure what you mean by "not using the sides"; does that 
mean the edge of the paper can't form one of the sides of the 
triangle, or that you can't assume the sides are parallel, or what? I 
think what I've done probably is allowed, regardless of the 

Origami books teach a standard way to make a 60-degree angle by 
folding, which you can use to accomplish the task. If you have two 
parallel lines and a perpendicular to them, form a third parallel 
midway between the first two, and then fold the perpendicular AB at an 
angle so that point A lies on the midline (A'), while the fold passes 
through point B. Then the fold BC will be at the desired angle:

            |\  \
            | \    \
            |  \       \
            |    \        \
            |     \           \
            |       \            /
            |        \          /
            |          \      /
            |           \    /
            |            \ /

From the fact that A'B = AB, you can prove easily that ABA' is an
equilateral triangle, so that angle ABC measures 30 degrees.

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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