# Kepler–Bouwkamp constant

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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 the inverse of the polygon circumscribing constant.

## Numerical value of the Kepler–Bouwkamp constant

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

$\prod_{k=3}^\infty \cos\left(\frac\pi k\right) = 0.1149420448\dots.$

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

$\prod_{k=3,5,7,11,13,17,\ldots} \cos\left(\frac\pi k\right) = 0.312832\ldots$

is obtained (sequence A131671 in OEIS).