A sphere is inscribed in a regular quadrangular prism, find the ratio of the total

A sphere is inscribed in a regular quadrangular prism, find the ratio of the total surface area of the prism to the total surface area of the sphere.

Since the sphere is inscribed in the correct prism, this prism is a cube.

Let the length of the cube be equal to X cm, then the area of its lateral faces is equal to X ^ 2 cm2.

Since the cube has six faces, then Sпов = 6 * X ^ 2.

The diameter of a sphere inscribed in a cube is equal to the length of an edge of the cube. D = X cm.

Then the area of the sphere is equal to: Sсф = π * D ^ 2 = π * X ^ 2.

Let’s define the area ratio.

Sпов / Sсф = 6 * X ^ 2 / π * X ^ 2 = 6 / π.