# The diagonal of the axial section of the cylinder is 48 cm. The angle between the diagonal and the generatrix

**The diagonal of the axial section of the cylinder is 48 cm. The angle between the diagonal and the generatrix of the cylinder is 60 °. Find the height of the cylinder, the radius of the cylinder, and the area of the lateral surface of the cylinder.**

A cylinder is a geometric body created by rotating a rectangle around one of its sides.

The axial section of a cylinder is a plane that passes through the axis of the cylinder.

The diagonal of the axial section, which forms and the diameter of the base, make up a right-angled triangle. The height of the cylinder is equal to the length of its generatrix. Therefore, to calculate its length, we apply the cosine theorem:

cos α = H / d;

H = d cos α, where:

H – height;

d is the diagonal of the axial section;

α is the angle between the diagonal of the axial section and the generatrix;

cos 60º = 1/2;

H = 48 1/2 = 24 cm.

To calculate the diameter of the base, we apply the Pythagorean theorem:

d ^ 2 = D ^ 2 + H ^ 2;

D ^ 2 = d ^ 2 – H ^ 2;

D ^ 2 = 48 ^ 2 – 24 ^ 2 = 2304 – 576 = 1728;

D = √1728 = 41.6 cm;

R = D / 2;

R = 41.6 / 2 = 20.8 cm.

The lateral surface area of the cylinder is calculated by the formula:

Sb.p. = 2πRH;

Sb.p. = 2 * 3.14 * 20.8 * 24 ≈ 3135 cm2.

Answer: The height of the cylinder is 24 cm, the radius is 20.8 cm, and the lateral surface area is 3135 cm2.