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=Bh By the Transitive Property of Equality, a formula for the volume of the cylinder can be written. V_C= V_P V_P=Bh ⇒ V_C=Bh 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=Bh 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.