The volume of a cylinder is calculated by multiplying the area of its base by its height.
If the radius of the base of a cylinder is r, the volume can be calculated by the following formula.
V=πr2h
Proof
Informal Justification
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.
VC=VP
Furthermore, the volume of a prism can be calculated by multiplying the area of its base by its height.
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=πr2 can be substituted in the above formula.
VC=BhsubstituteVC=πr2h
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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