Apothem of the pyramid is right. the triangular is 5 cm and the height is 4 cm. Find the S (side) of the pyramid.

Consider a right-angled triangular SOD. Leg OD, is the radius of the inscribed circle in the equilateral triangle of the base of the pyramid.

By the Pythagorean theorem OD ^ 2 = SD ^ 2 – SO ^ 2 = 25 – 16 = 9. OD = 3 cm.

Determine the side of the base, knowing the radius of the inscribed circle.

R = AC / 2 * √3.

3 = AC / 2 * √3.

AC = 6 * √3.

Let us find the area of the lateral surface, which is equal to half of the product of the base perimeter and the apothem.

Side = Ravs * SD / 2 = 3 * 6 * (√3) * 5/2 = 45 * √3.

Answer: Side = 45 * √3.