geometry

Suppose you have an equilateral triangle. The area of the triangle is exactly 1200 square centimetres. Now suppose you have twenty of those triangles. It's possible to assemble those twenty triangles into a closed three-dimensional shape, a regular polyhedron.

What would be the volume, in cubic centimetres, of the largest sphere that could fit inside the shape?

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  1. Please stop cheating at neopets!

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  2. The formula that you need for the radius of the inscribed sphere can be found in the link below, but deriving it would be rather difficult:

    http://en.wikipedia.org/wiki/Icosahedron

    Once you have the radius, compute the sphere volume in the usual way.

    The Wikipedia reference gives the sphere radius in terms of the triangle side length, a. In your case, the side length is
    a = [4/sqrt3 * A]^1/2 = 52.643 cm

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