In a regular triangular prism, the sides of the base are 6, the side edges are 7 cm. Find the volume of the prism.

The area of the lateral surface of the pyramid is calculated by the formula Sbok = 1/2 * P * A (P – base perimeter, A – apothem).
We calculate the perimeter of the base: 6 + 6 + 6 = 18 (cm).
Apothem is the height and median of the side face of the pyramid, the side face is an isosceles triangle.
By the Pythagorean theorem:
A = √ (7² – 3²) = √ (49 – 9) = √40 = 2√10.
Let’s find the area of the lateral surface of the pyramid:
Side = 1/2 * 18 * 2√10 = 18√10 (cm²).