Polygon Inscribing Constant
If a Triangle is inscribed in a Circle, another Circle inside the Triangle, a Squareinside the Circle, another Circle inside the Square, and so on,
Numerically,
where
is the Polygon Circumscribing Constant. Kasner and Newman's (1989)assertion that 
is incorrect.Let a convex Polygon be inscribed in a Circle and divided into Triangles from diagonals from oneVertex. The sum of the Radii of the Circles inscribed in theseTriangles is the same independent of the Vertex chosen (Johnson 1929, p. 193).
See also Polygon Circumscribing Constant
References
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/infprd/infprd.html 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://www.lacim.uqam.ca/piDATA/productcos.txt.
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