Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex . The sum of the radii
of the circles inscribed in these triangles
is the same independent of the polygon vertex chosen
(Johnson 1929, p. 193).
If a triangle is inscribed in a circle , another circle inside the triangle ,
a square inside the circle ,
another circle inside the square ,
and so on. Then the equation relating the inradius and
circumradius of a regular
polygon ,
(1)
gives the ratio of the radii of the final to initial circles as
(2)
Numerically,
(3)
(OEIS A085365 ), where polygon
circumscribing . This constant is termed the Kepler-Bouwkamp constant by Finch
(2003). Kasner and Newman's (1989) assertion that et al. (1986, p. 757).
See also Infinite Product ,
Nested Polygon ,
Polygon Circumscribing ,
Whirl
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References Finch, S. R. "Kepler-Bouwkamp Constant." §6.3 in Mathematical
Constants. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Kasner, E. and Newman, J. R.
Mathematics
and the Imagination. Pappas,
T. "Infinity & Limits." The
Joy of Mathematics. Plouffe, S. "Product(cos(Pi/n),n=3..infinity)." http://pi.lacim.uqam.ca/piDATA/productcos.txt . Prudnikov,
A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals
and Series, Vol. 1: Elementary Functions. Sloane, N. J. A. Sequence A085365
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Polygon Inscribing
Cite this as:
Weisstein, Eric W. "Polygon Inscribing."
From MathWorld https://mathworld.wolfram.com/PolygonInscribing.html
Subject classifications
is the corresponding constant for polygon
circumscribing. This constant is termed the Kepler-Bouwkamp constant by Finch
(2003). Kasner and Newman's (1989) assertion that 
is incorrect, as is the value of 0.8700... given by Prudnikov
et al. (1986, p. 757).
