Volume of a Cylinder

Consider a rectangular prism and a right cylinder that have the same base area and height.

In this case, the cross-sections of the prism and the cylinder are congruent to their bases. Therefore, their cross-sectional areas at every altitude are equal.

By the Cavalieri's Principle, two solids with the same height and the same cross-sectional area at every altitude have the same volume. Therefore, the volume of the cylinder is the same as the volume of the prism.

V_{C} = V_{P}

Furthermore, the volume of a prism can be calculated by multiplying the area of its base by its height.

V_{P} = B h

By the Transitive Property of Equality, a formula for the volume of the cylinder can be written.

{V_{C} = V_{P} V_{P} = B h \Rightarrow V_{C} = B h

Finally, not only is

B

the area of the base of the prism, but also the area of the base of the cylinder. Since the base of the cylinder is a circle, its area is the product of

π

and the square of its radius

r .

Therefore,

B = π r^{2}

can be substituted in the above formula.

V_{C} = B h substitute V_{C} = π r^{2} h

This formula applies to all cylinders because there is always a prism with the same base area and height. Also, by the Cavalieri's principle, this formula still holds true for oblique cylinders.

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Rule

Volume of a Cylinder

Proof

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