Find the volume of a regular triangular pyramid with side edges equal to 8 and inclined to the base plane

Find the volume of a regular triangular pyramid with side edges equal to 8 and inclined to the base plane at an angle of 60 degrees.

AO = AS * cos 60, then SO = AS * sin 60.
Let’s transform: AO = 8 * 0.5 = 4.
SO = 8 * √3 / 2;
SO = 4√3 is the h of the pyramid.
AO = 2/3 * AK, where AK – h.
AK = 3/2 * AO, substitute AK = 3/2 * 4 = 6.
By the Pythagorean theorem:
AK ² = a ² – (a / 2) ²;
a ² = 4/3 * AK ²;
a = 4√3.
Base area:
S = (a * h) / 2;
S = (4√3 * 6) / 2 = 12√3;
V = (S * H) / 3;
V = (12√3 * 4√3) / 3 = 48.
Answer: 48 cm ³.