Equilateral Triangles

The sides of an equilateral triangle are all the same length, and the angles are all the same size.

Manipulating Equilateral Triangles

Students can verify this by taking two equilateral triangles that are the same size, laying them one on top of the other, then rotating and flipping either triangle in as many ways as possible. They always line up perfectly.

Students can make beautiful and interesting geometric designs from equilateral triangles of various sizes and colors. By comparing and describing their designs they will discover numerous facts about triangles, hexagons, and symmetry.

An Important Question

How can you be sure that the triangle made from the folding procedure is actually an equilateral triangle?

One way, of course, is to make two of them, place them on top of one another, and rotate one of them until you are convinced that all of the sides and angles are equal. Or you could do various cutting or folding experiments on a single triangle to see if the sides and/or angles are equal.

Your results also will be dependent on how precisely you made the triangles, and how accurately you compare the lengths of sides and sizes of angles to see if they are the same. What if there are differences in lengths of sides or sizes of angles that are too small to be seen with the naked eye?

Another problem with experiments like these is that they tell you about the triangles that you have right in front of you, but they won't tell you anything about every triangle that you or anyone else could ever possibly make from this folding procedure. Look at the figure. Is there enough information there to be certain that the procedure only makes equilateral triangles?