The area of the base of a regular triangular pyramid is 9√3 cm ^ 2, and the apothem of the pyramid is 5 cm.

The area of the base of a regular triangular pyramid is 9√3 cm ^ 2, and the apothem of the pyramid is 5 cm. Find the area of the lateral surface of the pyramid.

An equilateral triangle lies at the base of a regular triangular pyramid. Its area is known by condition. Find the side of the base.
S main. = √3 / 4 * a²;
a² = S main. / √3 / 4 = 9√3 / √3 / 4 = 36;
a = √36 = 6 (cm) – the side of an equilateral triangle at the base of the pyramid.
The perimeter of the base is:
P = 3 * a = 18 (cm).
Apothem will be denoted by A.
We find the lateral surface area by the formula:
S side. = 1/2 * P * A = 1/2 * 18 * 5 = 45 cm².
Answer: the area of the lateral surface of the pyramid is 45 cm².