Volume of Cone and Cylinder

Picture a rectangle divided into two right triangles by a diagonal. How is the area of the right triangle formed by the diagonal related to the area of the rectangle? The area of any rectangle is the product of its width and length. For example, if a rectangle is 3 inches wide and 5 inches long, its area is 15 square inches (length times width). The figure below shows a rectangle "split" along a diagonal, demonstrating that the rectangle can be thought of as two equal right triangles joined together. The areas of rectangles and right triangles are proportional to one another: a rectangle has twice the area of the right triangle formed by its diagonal.

In a similar way, the volumes of a cone and a cylinder that have identical bases and heights are proportional. If a cone and a cylinder have bases (shown in color) with equal areas, and both have identical heights, then the volume of the cone is one-third the volume of the cylinder.

Imagine turning the cone in the figure upside down, with its point downward. If the cone were hollow with its top open, it could be filled with a liquid just like an ice cream cone. One would have to fill and pour the contents of the cone into the cylinder three times in order to fill up the cylinder.

The figure above also illustrates the terms height and radius for a cone and a cylinder. The base of the cone is a circle of radius r. The height of the cone is the length h of the straight line from the cone's tip to the center of its circular base. Both ends of a cylinder are circles, each of radius r. The height of the cylinder is the length h between the centers of the two ends.

The volume relationship between these cones and cylinders with equal bases and heights can be expressed mathematically. The volume of an object is the amount of space enclosed within it. For example, the volume of a cube is the area of one side times its height. The figure below shows a cube. The area of its base is indicated in color. Multiplying this (colored) area by the height L of the cube gives its volume. And since each dimension (length, width and height) of a cube is identical, its volume is L × L × L, or L ³, where L is the length of each side.

The same procedure can be applied to finding the volume of a cylinder. That is, the area of the base of the cylinder times the height of the cylinder gives its volume. The bases of the cylinder and cone shown previously are circles. The area of a circle is πr ², where r is the radius of the circle. Therefore, the volume V _cyl is given by the equation: V _cyl πr ²h (area of its circular base times its height) where r is the radius of the cylinder and h is its height. The volume of the cone (V _cone) is one-third that of a cylinder that has the same base and height: .

The cones and cylinders shown previously are right circular cones and right circular cylinders, which means that the central axis of each is perpendicular to the base. There are other types of cylinders and cones, and the proportions and equations that have been developed above also apply to these other types of cylinders and cones.

Philip Edward Koth with

William Arthur Atkins

Bibliography

Abbott, P. Geometry. New York: David Mckay Co., Inc., 1982.

Internet Resources

The Method of Archimedes. American Mathematical Society. <http://www.ams.org/new-in-math/cover/archimedes2.html>.