In a regular quadrangular pyramid, the height is 6 cm, and the lateral rib forms an angle of 30 degrees

| May 14, 2021 | education

In a regular quadrangular pyramid, the height is 6 cm, and the lateral rib forms an angle of 30 degrees with the base plane. Find: a) the volume of the pyramid, b) the height of a cylinder equal in size to the given pyramid, if the radius of the base is 4 cm (bodies are called equal if their volumes are equal).

Since the side ribs are inclined to the plane of the base at an angle of 300, then in the right-angled triangle MOC the angle OCM = 300.

Then tg30 = MO / CO.

CO = MO / tg30 = 6 / (1 / √3) = 6 * √3 cm.

Point O divides the AC diagonal in half, then AC = 2 * CO = 12 * √3 cm.

Let’s define the area of the square at the base of the pyramid through its diagonal.

Sb = АС ^ 2/2 = 144 * 3/2 = 216 cm2.

Then V = Sbn * MO / 3 = 216 * 6/3 = 432 cm2.

Let us determine the area of the base of an equal-sized cylinder.

Sc = n * R ^ 2 = 16 * n cm2.

Then Vcyl = Sc * h = 432.

h = 432/16 * n = 27 / n cm.

Answer: The volume of the pyramid is 432 cm2, the height of the cylinder is 27 / n cm.



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