Find the volume of a regular quadrangular pyramid, whose side faces are regular triangles, and the side edge is 6 cm.

At the base of the pyramid is a square with a side of 6 cm, then Sosn = AB * AB = 6 * 6 = 36 cm2.
Determine the length of the diagonal of the square at the base. AC = AB * √2 = 6 * √2 cm, then the segment AO = AC / 2 = 6 * √2 / 2 = 3 * √2 cm.
From a right-angled triangle AOD, we define, according to the Pythagorean theorem, the leg OD.
OD^2 = AD^2 – AO^2 = 36 – 18 = 18.
OD = √18 = 3 * √2 cm.
Let’s define the volume of the pyramid.
V = Sbase * OD / 3 = 36 * 3 * √2 / 3 = 36 * √2 cm3.
Answer: The volume of the pyramid is 36 * √2 cm3.