The height of a regular triangular pyramid is 4 cm, and the angle between the height

The height of a regular triangular pyramid is 4 cm, and the angle between the height and the side edge is 30 degrees. find the volume of the pyramid.

In a right-angled triangle AOD, we determine the size of the leg AO.

tg30 = ОА / ОD, then ОА = ОD * tan30 = 4 * (√3 / 3) cm.

The segment OA is the radius of the circumscribed circle about the equilateral triangle ABC, which is equal to: R = a / √3, where a is the side of the triangle ABC.

Then a = AB = BC = AC = AO * √3 = 4 * (√3 / 3) * √3 = 4 cm.

Determine the area of the base of the pyramid. Sbn = ВС ^ 2 * √3 / 4 = 16 * √3 / 4 = 4 * √3 cm2.

Let’s define the volume of the pyramid.

V = Saavs * OD / 3 = 4 * √3 * 4/3 = 16 * √3 / 3 cm3.

Answer: The volume of the pyramid is 16 * √3 / 3 cm3.