# 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. One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.