The ball is inscribed in the cylinder. The surface area of the ball is 48. Find the total surface area of the cylinder.

A ball can be inscribed in a cylinder provided that the height of the cylinder – h is equal to the diameter of its base – d.

The diameter of the inscribed sphere D will be equal to these values ​​D = d = h.

This implies the equality of the radii of the ball – R and the base of the cylinder – r.

Let us express the surface area of ​​the ball in terms of its radius, as well as the total surface area of ​​the cylinder in terms of the base radius.

S (w) = 4πR ^ 2;

The area of ​​the cylinder S (q) is the sum of the area of ​​the two bases 2 ∙ πr ^ 2 and the lateral surface h ∙ 2πr.

Considering that h = d = 2r, we get 2r ∙ 2πr = 4πr ^ 2.

S (q) = 2πr ^ 2 + 4πr ^ 2 = 6πr ^ 2.

Since R = r, S (w) / S (q) = 4/6 = 2/3.

Using this formula, we express the full surface of the cylinder through the surface of the ball: S (q) = (3/2) ∙ S (w); S (q) = (3/2) ∙ 48 = 72.

Answer: The total surface area of ​​the cylinder is 72.