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.