How to find the area of the lateral surface of a cylinder, the diagonal of the axial cut is 13 cm, and the height is 12 cm.

A cylinder is a solid created by rotating a rectangle around its side.

The axial section of a cylinder is a plane passing through its axis. It has the shape of a rectangle. Let’s designate it ABCD.

The diagonal of the axial section divides it into two, equal to each other, right-angled triangles.

Consider the triangle ΔABS.

Side AB is the height of the cylinder, as well as the leg of this triangle, and is 12 cm.

The AC side is the diagonal of the axial section, the hypotenuse of the triangle and is equal to 13 cm.

With this, we can find the diameter of the base of the cylinder, which is the leg of the BC. Let’s apply the Pythagorean theorem:

AC ^ 2 = AB ^ 2 + BC ^ 2;

BC ^ 2 = AC ^ 2 – AB ^ 2;

BC ^ 2 = 13 ^ 2 – 12 ^ 2 = 169 – 144 = 25;

BC = √25 = 5 cm.

Since the base radius is half the diameter, then:

r = d / 2;

r = 5/2 = 2.5 cm.

The lateral surface area of ​​the cylinder is:

Sb.p. = 2πrh;

Sb.p. = 2 · 3.14 · 2.5 · 12 = 188.4 cm2.

Answer: the area of ​​the lateral surface of the cylinder is 188.4 cm2.