A cube and a regular octahedron are carved out of of unit-radius wooden spheres . If the vertices are on the spheres, how do you prove that their volumes compare with that of the sphere, in the proportions 3√2:2:π?

1 Answer
Shwetank Mauria
Nov 16, 2016

The proportion is 2√3:1:π

Explanation:

Volume of a sphere of radius r is given by 43πr3. As radius of given sphere is unit, its volume will be 4π3.

Now let us consider a cube carved in unit sphere. It should appear as follows:
enter image source here
As the diameter of sphere is the longest diagonal of sphere, which is 2 here, 3s2=22 or s=√43=2√3 and volume of cube is (2√3)3=83√3.

A regular octahedron is a solid object made of eight equilateral triangles and appears as shown below. It is made of two tetrahedrons and volume of an octahedron of side a is given by √23a3.
enter image source here
Let us consider an octahedron in a sphere, so that when a sphere is divided into eight equal parts each part contains an equilateral triangle. Using Pythagoras theorem, the side of a tetrahedron will be given by a2=r2+r2=2r2 and a=r×√2.

Hence, volume of tetrahedron in a sphere of unit radius will be √23(√2)3=43,

Now we have to find ratio of volume of such cube, octahedron and sphere and it is

83√3:43:4π3

and multiplying each term by 34, we get

83√3×34:43×34:4π3×34

or 2√3:1:π

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