In a regular triangular pyramid, the side edge forms an angle of 30 ° with the base plane.

| September 24, 2021 | education

In a regular triangular pyramid, the side edge forms an angle of 30 ° with the base plane. Find the volume of the pyramid if its height is 8cm.

Let’s draw the height АН at the base of the pyramid, which is also the median of the triangle ABC.

The height of the MO of the pyramid, the side face of the MA and the segment AO form a right-angled triangle AMO, in which we determine the length of the leg OA.

tgMAO = MO / AO.

AO = MO / tg30 = 8 / (1 / √3) = 8 * √3 cm.

By the property of the medians of the triangle, they are divided at the point of intersection in the ratio of 2/1, then OH = AO / 2 = 8 * √3 / 2 = 4 * √3 cm.

Then AH = AO + OH = 12 * √3 cm.

The height of an equilateral triangle is: AH = BC * √3 / 2, then:

ВС = 2 * АH / √3 = 2 * 12 * √3 / √3 = 24 cm.

Determine the area of ​​the triangle ABC.

Sаvs = АН * ВС / 2 = 12 * √3 * 24/2 = 144 * √3 cm2.

Let’s define the volume of the pyramid.

V = Saavs * MO / 3 = 144 * √3 * 8/3 = 384 * √3 cm3.

Answer: The volume of the pyramid is 384 * √3 cm3.



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