A small sphere is centered at a corner of a large cube. What proportion of the surface of the sphere is inside of the cube?
Answer the same question with the cube replaced with a regular octahedron.
Possible answer:
I think every polyhedron has 4π steradians as the sum of the "angular defects" of all vertices.
Since cubes have 8 vertices, they have π/2 at each vertex and 1/8 of the sphere inside the cube.
Regular octahedrons have 6 vertices, and therefore have 4π/6 = 2π/3 steradians angle at each vertex.
So the regular octahedron has 1/6 of the sphere inside it.
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Posted by Larry
on 2025-02-28 13:14:06 |
Possible answer
