The base of the straight prism is a right-angled triangle with legs 5 and 12, surface area 390. find its height.

By the Pythagorean theorem, we find the hypotenuse of the base:

c ^ 2 = 5 ^ 2 + 12 ^ 2 = 25 +144 = 169;

c = √169 = 13.

The area of a right-angled triangle lying at the base of this prism is equal to half the product of the legs:

Sb = 0.5 * 5 * 12 = 30.

The total surface area is equal to the sum of the areas of the two bases and the lateral surface area:

S full = 2 * S main + S side.

The lateral surface area can be defined as the product of the perimeter of the base and the height of the prism:

Sside = P * h.

Hence:

h = (Sful – 2 * Sbn) / P = (390 – 2 * 30) / (5 + 12 + 13) = 330/30 = 11.