The diagonal of the axial section of the cylinder makes an angle γ with the generatrix. Find the volume of the cylinder

| August 5, 2021 | education

The diagonal of the axial section of the cylinder makes an angle γ with the generatrix. Find the volume of the cylinder, the lateral surface area, and the total surface area if the height of the cylinder is d.

In a right-angled triangle AВD tgY = AV / AP.

BP = AB / tgY = d / tgY.

AD is the diameter of the circle at the base of the cylinder, then R = AD / 2 = d / 2 * tgY = d * ctgY / 2 cm.

Determine the volume of the cylinder.

V = π * R ^ 2 * AB = π * d ^ 2 * d * ctg2Y / 4 = π * d ^ 3 * ctg2Y / 4 cm3.

Let us determine the area of the lateral surface.

Side = 2 * π * R * AB = 2 * π * (d * ctgY / 2) * d = π * d ^ 2 * ctgY cm2.

Let us determine the total surface area of the cylinder.

Spov = π * R ^ 2 + Sside = π * d ^ 2 * ctg2Y / 4 + π * d ^ 2 * ctgY = (π * d ^ 2 * ctgY) * ((1 * ctgY / 4) + 1) cm2 …

Answer: V = d ^ 3 * ctg2Y / 4 cm3, Side = π * d ^ 2 * ctgY cm2, Spov = (π * d ^ 2 * ctgY) * ((1 * ctgY / 4) + 1) cm2.



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