In a regular quadrangular pyramid, the lateral rib is inclined to the plane of the base at an angle of 45 degrees, the radius

In a regular quadrangular pyramid, the lateral rib is inclined to the plane of the base at an angle of 45 degrees, the radius of the circumscribed circle near the base is 4 cm. Find the volume.

Since the pyramid is correct, ABCD is a square, and therefore, the radius of the circle described around it is half the diagonal of the square.

Then AC = BD = 2 * R = 2 * 4 = 8 cm.

Since the side ribs are inclined to the base plane at an angle of 45, the SOC triangle is rectangular and isosceles, SO = OC = R = 4 cm.

Determine the area of the base of the pyramid.

Sb = D ^ 2/2 = AC ^ 2/2 = 64/2 = 32 cm2.

Then the volume of the pyramid is: V = Sosn * SO / 3 = 32 * 4/3 = 128/3 = 42 (2/3) cm3.