# Hexagonal pyramid, height = 16, apothem = 20. Find the lateral surface area

Consider a right-angled triangle РOН, in which the angle O is a straight line, as the height lowered to the base, the hypotenuse of РН is the apothem of the pyramid. By the Pythagorean theorem, we find the leg OH.

HE: 2 = PH: 2 – OR: 2 = 20: 2 – 16: 2 = 400 – 256 = 144.

OH = 12 cm.

Since a regular hexagon is located at the base, then all the triangles formed by the diagonals are equilateral and the angles in them are 60.

Consider the triangle AOK. In which the height is OH = 12 cm.

Then in a right-angled triangle AON, the hypotenuse OA is equal to the side of the hexagon and is equal to:

ОА = ОН / SinA = 12 / Sin60 = 12 / (√3 / 2) = 24 / √3 = 8 * √3.

Then the area of the side surface of the pyramid is equal to:

S = P * h / 2, where P is the perimeter of the base, h is the height of the side face.

S = 6 * AO * PН / 2 = 6 * 8 * 20 * √3 / 2 = 480 * √3 cm2.

Answer: Side = 480 * √3 cm2.