# The apothem of a regular triangular pyramid is 4 cm, and the dihedral angle at the base is 60

**The apothem of a regular triangular pyramid is 4 cm, and the dihedral angle at the base is 60 degrees. Find the volume of the pyramid.**

The dihedral angle at the base is the linear angle DHA between the apothem and the height AH of the triangle at the base of the pyramid. The triangle DOH is rectangular, the angle ODH = (90 – 60) = 30. Then the leg OH lies opposite the angle 30, which means OH = DH / 2 = 4/2 = 2 cm.

Then DO ^ 2 = DH ^ 2 – OH ^ 2 = 16 – 4 = 12. DO = 2 * √3 cm

Since ABC is equilateral, the median AH at point O is divided in the ratio OA / OH = 2/1.

Then OA = 2 * OH = 2 * 2 = 4 cm, AH = OA + OH = 4 + 2 = 6 cm.

In a right-angled triangle AHC CH = AC / 2. Let CH = X cm, then AC = 2 * X cm.

By the Pythagorean theorem, 4 * X ^ 2 = AH ^ 2 + X ^ 2.

3 * X ^ 2 = 36.

X ^ 2 = 36/3 = 12.

X = CH = 2 * √3 cm, then CB = 2 * CH = 4 * √3 cm.

Determine the area of the base of the pyramid. Sbn = CВ * АH / 2 = 4 * √3 * 6/2 = 12 * √3 cm2.

Let’s define the volume of the pyramid.

V = Sbase * DO / 3 = 12 * √3 * 2 * √3 / 3 = 24 cm3.