At a distance m from the cylinder axis, a section is drawn parallel to the axis and cutting off the alpha arc from the base circle.

| June 1, 2021 | education

At a distance m from the cylinder axis, a section is drawn parallel to the axis and cutting off the alpha arc from the base circle. The diagonal of the section intersects the generatrix of the cylinder at an angle of beta. Find the volume of the cylinder.

Let’s build the radii РA and OB. The AOB triangle is isosceles. The height OH is also the bisector and median of the AOB triangle. By condition, OH = m see.

Arc AB = α, then the central angle AOB = α0, and then the angle AOH = (α / 2).

In a right-angled triangle, tg (AOH) = AH / OH.

AH = OH * tan (α / 2) = m * tan (α / 2), then AB = 2 * m * tan (α / 2).

Cos (α / 2) = OH / OA.

OA = OH / Cos (α / 2) = m / Cos (α / 2).

Then Sop = π * ОА2 = π * m2 / Cos2 (α / 2).

A right-angled triangle ABC tgβ = AB / BC.

BC = AB / tanβ = 2 * m * tan (α / 2) / tanβ.

Then V = Sosn * BC = (π * m2 / Cos2 (α / 2)) * (2 * m * tan (α / 2) / tanβ).

Answer: The volume of the cylinder is: (π * m2 / Cos2 (α / 2)) * (2 * m * tg (α / 2) / tgβ).



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