How do the areas of spheres inscribed in a cube and described around the same cube compare?

The diameter of the sphere circumscribed about the cube is equal to the diagonal of the cube, the diameter of the sphere inscribed in the cube is equal to the length of the edge of the cube.

The area of ​​the sphere is determined by the formula: S = 4πR ^ 2 = πD ^ 2.

Let the edge of the cube be equal to a, then the diameter of the inscribed sphere d = a, and its area:

S1 = πd ^ 2 = πa ^ 2.

The square of the diagonal of a cube is equal to the sum of the squares of its three dimensions:

D ^ 2 = a ^ 2 + a ^ 2 + a ^ 2 = 3a ^ 2;

D = a√3.

The area of ​​a sphere described around a given cube:

S2 = πD ^ 2 = π * (a√3) ^ 2 = 3πa ^ 2.

We can find the ratio of the areas of the circumscribed and inscribed spheres:

S2 / S1 = 3πa ^ 2 / πa ^ 2 = 3.

Consequently, the area of ​​a sphere described about a cube is three times larger than the area of ​​a sphere inscribed in the same cube.