Volume of a cylindrical wedge

Geometry Level 2

A wedge is cut out of a cylinder of radius 25 cm 25 \text{cm} . The cutting plane passes through the center of the cylinder base and makes an angle of 3 0 ∘ 30^{\circ} with it. Find the volume of the wedge in cubic centimeters.


The answer is 6014.065.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Tom Engelsman
May 9, 2018

Let the wedge plane in question be modeled as z = tan ⁡ ( π 6 ) ⋅ y = y 3 z = \tan(\frac{\pi}{6}) \cdot y = \frac{y}{\sqrt{3}} , which the volume can be computed via the following triple integration:

V = 2 ⋅ ∫ 0 25 ∫ 0 2 5 2 − x 2 ∫ 0 y 3 d z d y d x = 31250 3 3 . V = 2 \cdot \int_{0}^{25} \int_{0}^{\sqrt{25^2 - x^2}} \int_{0}^{\frac{y}{\sqrt{3}}} dz dy dx = \boxed{\frac{31250}{3\sqrt{3}}}.

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