The height of a regular triangular pyramid is 2 √3 cm. Calculate the volume of the

| June 21, 2021 | education

The height of a regular triangular pyramid is 2 √3 cm. Calculate the volume of the pyramid if the side faces form an angle of 45 degrees with the base plane.

Since all the side faces of the pyramid are inclined at the same angle, the apex of the pyramid is projected to the center of the inscribed circle. Since the base of the pyramid is a regular triangle, point O is the intersection point of medians, heights and bisectors.

The triangle DOH is rectangular and isosceles, then OH = OD = 2 * √3 cm.By the property of the medians, AO = 2 * OH = 4 * √3 cm, then AH = 6 * √3 cm.

AH = BC * √3 / 2.

ВС = 2 * АH / √3 = 2 * 6 * √3 / √3 = 12 cm.

Sаvs = ВС * АН / 2 = 12 * 6 * √3 / 2 = 36 * √3 cm2.

Then V = Savs * OD / 3 = 36 * √3 * 2 * √3 / 3 = 72 cm3.

Answer: The volume of the pyramid is 72 cm3.



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