The height of a regular triangular pyramid is 12 cm. The dihedral angle at the base is 30 degrees. Find the volume of the pyramid.

| March 26, 2021 | education

The dihedral angle at the base is the angle АНD between the height АН of the base of the triangle and the height DH of the side face of ВСD. Then, in a right-angled triangle DOH, the leg OD lies opposite the angle 30, and therefore DH = 2 * OD = 2 * 12 = 24 cm.

Then, by the Pythagorean theorem, OH ^ 2 = DH ^ 2 – OD ^ 2 = 576 = 144 = 432.

OH = 12 * √3 cm.

The OH segment is the radius of the circle inscribed in the ABC triangle, which is equal to:

OH = BC * √3 / 6, then BC = 6 * OH / √3 = 6 * 12 * √3 / √3 = 72 cm.

Determine the area of ​​the triangle at the base of the pyramid.

Sbn = ВС2 * √3 / 4 = 1296 * √3 cm2.

Let’s define the volume of the pyramid.

V = Sbase * OD / 3 = 1296 * √3 * 2/3 = 864 * √3 cm3.

Answer: The volume of the pyramid is 864 * √3 cm3.



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