Folding a Dollar Bill into a Pyramid

This activity can be done with no preparation at all, in the classroom or as an amusement anywhere, anytime. All you need is a dollar bill! A crisp, new bill is best, and not only will the trick amuse your audience, but there is some fascinating mathematics in it. It would serve well as an introduction to 3-dimensional solids in a Geometry class, but will amuse students at all grade levels. It is actually the easiest way to create a 3-D model of a pyramid!

This activity can be done with no preparation at all, in the classroom or as an amusement anywhere, anytime. All you need is a dollar bill! A crisp, new bill is best, and not only will the trick amuse your audience, but there is some fascinating mathematics in it. It would serve well as an introduction to 3-dimensional solids in a Geometry class, but will amuse students at all grade levels. It is actually the easiest way to create a 3-D model of a pyramid.

In this activity, you begin with a dollar bill . .

...

... and end up with a tetrahedron, using a lot of math terminology along the way!

And on one of the faces of the dollar bill tetrahedron, you will see a drawing of a pyramid!

So, here are the steps to follow:

Take a dollar bill, with the side showing George Washington facing up as shown below, this time with the vertices labeled:

Fold line DC across the midline of rectangle ABCD as shown. Make a crisp crease on the midsegment, EF:

Fold the upper half CDEF back to it's original position. Then fold point D to the midline EF:

Fold triangle GAD across line GD:

Fold triangle GAH across line AH:

Fold triangle ACG across line AG:

Open up the figure. It will form a tetrahedron, as shown below:

... and on one of the faces of the dollar bill tetrahedron, you will see a drawing of a pyramid!

Besides serving as a good example of a pyramid, it is also a Tetrahedron; a pyramid with all faces and base congruent equilateral triangles. There are a number of interesting things that you will want to point out to your students. One face of the pyramid (the "open" face) is formed by 2 congruent 30/60/90 triangles. That is, each triangle has angles of 30 degrees, 60 degrees and 90 degrees. These 2 triangles have the numeral 1 on them, in perfect reflection symmetry.

It is interesting to explain to the class that this will not work with a piece of paper of any other ratio of dimensions (length and width). Clearly the dollar bill, and all of our paper currency, was designed in this exact ratio. And what is this ratio? If we call the shorter dimension of the right triangle 1 unit then the longer leg of the 30/60/90 triangle is 1 times the square root of 3, and the hypotenuse is 2. Therefore, the equilateral triangle has sides 2 units long, and the ratio of the length of a dollar bill to its width is the square root of 3 to 4.

If time permits, it is interesting and instructive to give students a variety of slips of paper, each with different proportions, and ask them to repeat the steps that they used to fold the dollar bill, with their paper rectangle. If none of the paper rectangles happens to be the same proportion as the dollar bill, the class will see that although each student has done the same steps, their piece of paper did not work as well as the dollar-bill shape; there will be either extra or insufficient paper in the last step.

It seems that the Treasury Department of the United States of America must have a high regard for mathematics, to have chosen this particular proportion!
We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others. Blaise Pascal (1623-1662)