Kepler–Bouwkamp constant

A sequence of inscribed polygons and circles

In plane geometry, the Kepler–Bouwkamp constant (or polygon inscribing constant) is obtained as a limit of the following sequence. Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth. The radius of the limiting circle is called the Kepler–Bouwkamp constant (Finch, 2003). It is named after Johannes Kepler and Christoffel Bouwkamp [de], and is the inverse of the polygon circumscribing constant.

Numerical valueEdit

The decimal expansion of the Kepler–Bouwkamp constant is (sequence A085365 in the OEIS)

The natural logarithm of the Kepler-Bouwkamp constant is given by

where   is the Riemann zeta function.

If the product is taken over the odd primes, the constant


is obtained (sequence A131671 in the OEIS).


  • Finch, S. R. (2003). Mathematical Constants. Cambridge University Press. MR 2003519.

Further readingEdit

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