The Area and Volume of Cones

A cone, according to MathWorld, is "a pyramid with a circular cross section." A cone has a circular base, or face. A common example of a cone is the ice-cream cone. The volume of a cone is a measure of how much it contains, and the surface area is how much area it would cover if were to be flattened out.

Volume of Cones

The formula to determine the volume of a cone is V = 1/3 x A x H, where A is the area of the base, and H is the height of the cone. Because the base of a cone is a circle, the area of the base is calculated using the formula A = pi x r x r or A = 22/7 x r x r, where r is the radius of the base. For example, a cone that is 4 inches high with a base that is 3 inches in diameter would have a volume of 1/3 x pi x 1.5 x 1.5 x 4 = approximately 10.5 cubic inches.

Right Circular Cones

Most cones are right circular cones. That is, they have circular bases and, if the vertex is pointed up, it is directly over the center of the base. A party hat is an example of a right circular cone.

Surface Area of Cones

The surface area of a cone cannot be determined without more information about its shape. For right circular cones, the surface area is A = pi x [(r x r) + (h x h)]^0.5. So, for our example above, the surface area would be A = 22/7 x [(1.5 x 1.5) + (3 x 3)]^0.5 = about 10.5 square inches. (The fact that the surface area is equals the volume is in this example is coincidental.)

References

• Mathworld
• Math for morons