Find the volume of a regular quadrangular pyramid, the base diagonal is 8√2, and the apothem is 5 cm.

We start with a short entry given:

SABCD – regular quadrangular pyramid,

AC = 8√2 cm, SE = 5 cm,

SE – apothem.

We are looking for the volume of the pyramid.

OC = 1/2 * AC = 1/2 * 8√2 = 4√2 cm.

From the OEC triangle:

OE is the middle line of the ADC triangle.

sin 45 ° = OE / OC; OE = OC / √2 = 4√2 / √2 = 4 cm.

SO is the height of the pyramid.

Apply the Pythagorean theorem to the SOE triangle:

SO ^ 2 = SE ^ 2 – OE ^ 2 = 5 ^ 2 – 4 ^ 2 = 25 – 16 = 9;

SO = 3 cm.

AD = 2 * OE = 2 * 4 = 8 cm.

It remains to find the volume:

V = 1/3 * SO * Sabcd = 1/3 * 3 * AD ^ 2 = 8 ^ 2 = 64 cm ^ 3.