In a regular quadrangular pyramid, the side of the base is 10 cm, and the side edge is 13 cm. Find the volume of the pyramid.

Determine the length of the AC diagonal at the base of the pyramid.

Since there is a square at the base of the pyramid, its diagonal is: AC = AB * √2 = 10 * √2 cm.

Point O divides the diagonals of the square in half, then OS = AC / 2 = 10 * √2 / 2 = 5 * √2 cm.

Determine the length of the height of the pyramid, from the right-angled triangle POC. By the Pythagorean theorem, PO ^ 2 = PC ^ 2 – OC ^ 2 = 169 – 50 = 119.

PO = √119 cm.

Determine the area of the base of the pyramid.

Sbn = AB ^ 2 = 10 ^ 2 = 100 cm2.

Then V = Sbase * PO / 3 = 100 * √119 / 3 cm3.

Answer: The volume of the pyramid is 100 * √119 / 3 cm3.