In a regular triangular pyramid, the apothem is equal to L and forms an angle a with the height of the pyramid.

| January 14, 2021 | education

In a regular triangular pyramid, the apothem is equal to L and forms an angle a with the height of the pyramid. Find the volume of the pyramid.

In a regular pyramid, point O is the projection of the top D of the pyramid and the center of ABC inscribed and described around the base.
In a right-angled triangle, DOН SinDСН = OH / DН.
OH = DN * SinDСH = L * Cosα.
CosDСН = OD / DН.
OD = DН * CosDSN = L * Sinα.
By the property of the medians of the triangle, the length of the segment OC = 2 * OH = 2 * L * Sinα.
CH = OH + OС = 3 * L * Sinα.
In an equilateral triangle ABC CH = AB * √3 / 2.
AB = 2 * CH / √3 = 6 * L * Sinα / √3 = 2 * √3 * L * Sinα.
Then Savs = AB * CH / 2 = (2 * √3 * L * Sinα) * (3 * L * Sinα) / 2 = 3 * √3 * L2 * Sin2α.
Then Vpir = Saws * DO / 3 = (3 * √3 * L2 * Sin2α) * L * Cosα / 3 = √3 * L3 * Sin2α * Cosα cm3.
Answer: The volume of the pyramid is √3 * L3 * Sin2α * Cosα cm3.



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