The generatrix of the cone is inclined to the plane of the base at an angle of 30 degrees, the radius

| September 16, 2021 | education

The generatrix of the cone is inclined to the plane of the base at an angle of 30 degrees, the radius of the base is 3 dm. Find the volume of the cone and its area to the lateral surface.

From the right-angled triangle ABO, we determine the length of the OB leg and the hypotenuse AB.

tgBAO = BO / AO.

BO = AO * tg30 = 3 * √3 / 3 = √3 dm.

Then, by the Pythagorean theorem, AB ^ 2 = AO ^ 2 + BO ^ 2 = 9 + 3 = 12.

AB = 2 * √3 dm.

Determine the area of the base of the cone.

Sb = π * R2 = 9 * π cm2.

Then V = Sosn * BО / 3 = 9 * π * √3 / 3 = 3 * √3 * π dm3.

Let us determine the area of the lateral surface of the cone.

Side = π * R * AO = π * 3 * 2 * √3 = 6 * √3 * π cm2.

Answer: The volume of the cone is 3 * √3 * π dm3, the lateral surface area is 6 * √3 * π cm2.



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