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 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).
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. Cambridge, England: Cambridge University Press, pp. 428-429,
2003. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
Boston, MA: Houghton Mifflin, 1929. Kasner, E. and Newman, J. R.
Mathematics
and the Imagination. Redmond, WA: Microsoft Press, pp. 311-312, 1989. Pappas,
T. "Infinity & Limits." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 180,
1989. 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. New York: Gordon & Breach,
1986. 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 --A Wolfram Web Resource. https://mathworld.wolfram.com/PolygonInscribing.html
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