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From the Fun Fact files, here is a Fun Fact at the Medium level:

Reuleaux Wheel

Figure 1
Figure 1

A Reuleaux Triangle is a plump triangle with rounded edges, formed in the following way: take the three points at the corners of an equilateral triangle, and connect each pair of points by a circular arc centered at the remaining point.

This triangle has some amazing properties. It is constant-width, meaning that it will hug parallel lines as it rolls. By rotating the centroid of the Reuleaux triangle appropriately, the figure can be made to trace out a square, perfect except for slightly rounded corners!

This idea has formed the basis of a drill that will carve out squares!

And, what do you think the ratio of its circumference to its width is?

Amazing fact: it is PI!

Presentation Suggestions:
Have students think about why this figure is constant width.

The Math Behind the Fact:
There are many other convex, constant-width figures, such as the circle and various Reuleaux polygons, and they all satisfy the same ratio of circumference to width!

How to Cite this Page:
Su, Francis E., et al. "Reuleaux Wheel." Math Fun Facts. <>.

Subjects:    geometry, calculus, analysis
Level:    Medium
Fun Fact suggested by:   Michael Moody
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