The height of a regular triangular pyramid is equal to 2√3 cm and the side face forms an angle

| March 23, 2021 | education

The height of a regular triangular pyramid is equal to 2√3 cm and the side face forms an angle of 60 degrees with the base of the pyramid. Find the volume of the pyramid.

The angle between the side face and the base of the pyramid is the linear angle OHD between the heights AH and DH, then in the right-angled triangle OHD we determine the length of the leg OH.

tg60 = ОD / ОН, then ОH = ОD / tg60 = 2 * √3 / √3 = 2 cm.

The segment OH is the radius of the inscribed circle in the equilateral triangle ABC, which is equal to: R = a * √3 / 6, where a is the side of the triangle ABC.

AB = BC = AC = a = OH * 6 / √3 = 2 * 6 / √3 = 4 * √3 cm.

Determine the area of ​​the base of the pyramid. Sbn = ВС ^ 2 * √3 / 4 = 16 * 3 * √3 / 4 = 12 * √3 cm2.

Let’s define the volume of the pyramid.

V = Savs * OD / 3 = 12 * √3 * 2 * √3 / 3 = 24 cm3.

Answer: The volume of the pyramid is 24 cm3.



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