The apothem of a regular quadrangular pyramid is 6, and the angle between

| July 15, 2021 | education

The apothem of a regular quadrangular pyramid is 6, and the angle between the planes of the base and the side face is 30 °. Find the volume of the pyramid.

Let’s draw the diagonals of the square at the base of the pyramid and connect the point O with the exact H apothem. the linear angle KНO is equal to the two-sided angle between the side face and the base of the pyramid, the angle KНO = 30. Then in a right-angled triangle KOH the leg KO lies opposite the angle 30, and therefore KO = KН / 2 = 6/2 = 3 cm.

CosKHO = OH / KH.

OH = KH * CosKHO = 3 * Cos30 = 3 * √3 / 2.

Since OH is the middle line of the ACD triangle, then AD = 2 * OH = 2 * 3 * √3 / 2 = 3 * √3 cm.

Let’s determine the area of the base of the pyramid.

Sbn = АD ^ 2 = 9 * 3 = 27 cm2.

Then V = Sosn * KO / 3 = 27 * 3/3 = 27 cm3.

Answer: The volume of the pyramid is 27 cm3.



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