The area of the diagonal section of a regular quadrangular pyramid is 36 and its height is 9. Find the volume of the pyramid.

FABCD – regular quadrangular pyramid;

triangle AFC – diagonal section;

S triangle AFC = 36 = 1/2 AC * FO;

FO – pyramid height = 9;

36 = 1/2 * AC * 9 → AC = 8;

At the base of the pyramid is a regular quadrangle, that is, a square with a diagonal AC = 8 and a side = a;

According to Comrade Pythagoras from tr. ADC: 2a ^ 2 = 64 → a = 4√2;

S base of the pyramid = Sq = (4√2) ^ 2 = 32;

Vpyramid = 1/3 * S base * height = 1/2 * 32 * 9 = 144 units2

Answer: Vpyramids = 144 units2