The reuleaux triangle is formed by the intersection of three identical circles passing through the centers of each other. Not only is it a shape with constant width (so that you can use it as a "wheel"), it also has real life applications. Here is how the reuleaux triangle is applied to drill a square hole.
If the reuleaux triangle is simply rotated about the circumcenter, of course it traces a circle. However, if the circumcenter follows a circular path, and the reuleaux triangle spins at a certain speed, the trace actually looks like a square!
On this worksheet, you can explore how the sizes of the objects affect the trace of the reuleaux triangle. Also, if the triangle spins in a different speed with its center, do you still get a square shape?
Here are several more questions for you.
Is it possible to create a shape that resembles a square even more?
Does the path of the circumcenter have to be circular?
How does this help making a square drill bit?
Is it practical to have such a non-circular path in a drill bit?