In a cylinder with a radius of 6cm, the diagonal of the axial section is 13cm. Find the total surface area of the cylinder.

Find the diameter of the cylinder:

D = 2r = 2 * 6 = 12 cm.

The diagonal of the axial section, the diameter and the height of the cylinder together form a right-angled triangle, in which the diagonal is the hypotenuse. We need to find the height of the cylinder. By the Pythagorean theorem

H = √ (a² – D²), where a is the diagonal of the axial section of the cylinder.

Find the height:

H = √ (13² – 12²) = √ (169 – 144) = √25 = 5 cm.

Let us find the total surface area of the cylinder:

S = 2πr² + 2πrH = 2πr * (r + H) = 2 * 6π * (6 + 5) = 12π * 11 = 132π = 414.69 cm².

Answer: The total surface area of the cylinder is approximately 414.69 cm².