Reuleaux Triangle

A Curve of Constant Width constructed by drawing arcs from each Vertex of an Equilateral Triangle between the other two Vertices. It is the basis for the Harry Watt square drill bit. It has the smallest Area for a given width of any Curve of Constant Width.

The Area of each meniscus-shaped portion is

$$A={\textstyle{1\over 6}}\pi r^2-{\textstyle{1\over 2}}r\left... ...2} r}\right)=\left({{\pi\over 6}-{\sqrt{3}\over 4}}\right)r^2,$$ (1)

where we have subtracted the Area of the wedge from that of the Equilateral Triangle. The total Area is then

$$A=3\left({{\pi\over 6}-{\sqrt{3}\over 4}}\right)r^2 + {\sqrt{3}\over 4}r^2 = {\pi-\sqrt{3}\over 2}\, r^2.$$ (2)

When rotated in a square, the fractional Area covered is

$$A_{\rm covered}=2\sqrt{3}+{\textstyle{1\over 6}}\pi-3 = 0.9877003907\ldots.$$ (3)

The center does not stay fixed as the Triangle is rotated, but moves along a curve composed of four arcs of an Ellipse (Wagon 1991).

See also Curve of Constant Width, Flower of Life, Piecewise Circular Curve, Reuleaux Polygon

References

Bogomolny, A. ``Shapes of Constant Width.'' http://www.cut-the-knot.com/do_you_know/cwidth.html.

Eppstein, D. ``Reuleaux Triangles.'' http://www.ics.uci.edu/~eppstein/junkyard/reuleaux.html.

Reuleaux, F. The Kinematics of Machinery. New York: Dover, 1963.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 52-54 and 381-383, 1991.

Yaglom, I. M. and Boltyansky, B. G. Convex Shapes. Moscow: Nauka, 1951.