Reuleaux Polygons: Curves of Constant Width in Geometry

What Are Reuleaux Polygons?

Reuleaux polygons are shapes formed by the intersection of multiple circular arcs centered at the vertices of a regular polygon. These shapes have a constant width, meaning that the distance between any two parallel supporting lines is always the same, regardless of their orientation.

The simplest example is the Reuleaux triangle, which can be thought of as an equilateral triangle with each side replaced by an arc from a circle centered at the opposite vertex.

Properties and Applications

Reuleaux polygons have several interesting properties. For instance, they can roll smoothly between two parallel lines without any gaps or overlaps, similar to circular disks. This property makes them useful in various engineering applications.

Another intriguing aspect is that the perimeter of a Reuleaux polygon is always pi times its width, just like a circle.

Construction and Visualization

The construction of a Reuleaux polygon involves taking an odd-sided regular polygon (3, 5, 7, 9, or 11 sides) and replacing each side with the arc of a circle centered at the opposite vertex. The radius of these circles is equal to the length of the side of the polygon.

By rotating a Reuleaux polygon within a square, one can observe that it maintains constant width throughout its motion, which is not true for most other shapes.

Area and Perimeter Calculations

The area of a Reuleaux polygon can be calculated using the formula: A = (1/2) * w^2 * (n * arcsin(1/n) - (n-2)), where w is the width and n is the number of sides.

The perimeter of any Reuleaux polygon is always pi times its width, a result that can be derived from the properties of circular arcs.

Frequently asked questions

Try it live

Everything above runs in your browser — open Reuleaux Polygons and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.