Shapes of constant width
A circle is a shape with constant radius. This shape has a constant diameter. This idea can be used to drill (almost) square holes and make coins which work in slot machines.
http://www.cut-the-knot.org/do_you_know/cwidth.shtml
http://web.mat.bham.ac.uk/C.J.Sangwin/howroundcom/roundness/applications.html