In a straight triangular prism, the sides of the base are 6, 8, 10. The total surface area is 240.

In a straight triangular prism, the sides of the base are 6, 8, 10. The total surface area is 240. Find the height of the prism.

Total surface area of a triangular prism: Sп = Sbp + 2 * Sо.

The area of the lateral surface of the prism: Sbp = P * h, where P is the perimeter of the base, h is the height.

Find P, P = 6 + 8 + 10 = 24.

Base area according to Heron’s formula: Sо = √ (p * (p – a) * (p – b) * (p – c)), where p = P / 2 = 12;

Sо = √ (12 * (12 – 6) * (12 – 8) * (12 – 10)) = √ (12 * 48) = √576 = 24.

Knowing the area of the base, we find the area of the lateral surface:

Sbp = Spp – 2 * Sо = 240 – 2 * 24 = 192.

We calculate the height h = Sbp / P = 192/24 = 8.