The side face of a regular quadrangular pyramid is inclined to the base plane at an angle α. The segment

| September 3, 2021 | education

The side face of a regular quadrangular pyramid is inclined to the base plane at an angle α. The segment that connects the middle of the height of the pyramid and the middle of the apothem is equal to a. Find the volume of the pyramid.

Since point H is the middle of the height OK, and point P is the middle of the apothem KM, then the segment HP is the middle line of the right-angled triangle KOM, then OM = 2 * HP = 2 * a cm.

Since the pyramid is correct, there is a square at its base. Point M is the middle of the SD side, since the KM apothem is the height and median of the lateral face. Then AB = BC = CD = AC = 2 * OC = 4 * a cm.

Then Sosn = AB2 = 16 * a ^ 2.

In a right-angled triangle KOM, tgα = OK = OM = h / OM.

h = ОМ * tgα = 2 * a * tgα.

Then V = Ssc * h / 3 = 16 * a ^ 2 * 2 * a * tanα / 3 = 32 * a ^ 3 * 2 * tanα / 3 cm3.

Answer: The volume of the pyramid is 32 * a ^ 3 * 2 * tgα / 3 cm3.



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