Archimedean circle

In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and r denotes the radius of any of the inner half circles, then the radius ρ of such an Archimedean circle is given by

${\displaystyle \rho ={\frac {1}{2}}r\left(1-r\right),}$

There are over fifty different known ways to construct Archimedean circles.