What is the radius of these spheres?

Question

Begin with one red sphere of unit radius. On its surface evenly distribute twenty congruent blue spheres. Each of the blue spheres is externally tangent to the red sphere and to exactly three of the other blue spheres. What is the radius of a blue sphere?

Answer 1

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As the centers of the twenty spheres are at the vertices of a regular dodecahedron, their radii r satisfy the equation

1+r = r/2 (√¯15 + √¯3)

the answer is

r = 2/(√¯15 + √¯3 - 2) = 0.554779765

Source(s): http://en.wikipedia.org/wiki/Dodecahedron