Volume of a Cone

Which has a greater volume, a pyramid, cone, or cylinder if the figures have bases with the same "diameter" (i.e., distance across the base) and the same altitude? To find out, compare pyramids, cylinders, and cones that have bases with equal diameters and the same altitude.

Here are three figures that have the same dimensions—cylinder, a right regular hexagonal pyramid, and a right circular cone. Which figure appears to have a greater volume?

It seems obvious that the volume of the cylinder is greater than the other two figures. That’s because the pyramid and cone taper off to a single point, while the cylinder’s sides stay the same width.

Determining whether the pyramid or the cone has a greater volume is not so obvious. If you look at the bases of each figure you see that the apothem of the hexagon is congruent to the radius of the circle. You can see the relative size of the two bases by superimposing one onto the other.

From the diagram you can see that the hexagon is slightly larger in area than the circle. So it follows that the volume of the right hexagonal regular pyramid would be greater than the volume of a right circular cone. And indeed it is, but only because the area of the base of the hexagon is slightly greater than the area of the base of the circular cone.

The formula for finding the volume of each figure is virtually identical. Both formulas follow the same basic form:

V = \frac{1}{3}Bh

Since the base of a circular cone is, by definition, a circle, you can substitute the area of a circle, \pi r^2 for the base of the figure. This is expressed as a volume postulate for cones.

Volume of a Right Circular Cone

Given a right circular cone with height h and a base that has radius r:

V &= \frac{1}{3} Bh \\ &= \frac{1}{3} \pi r^2 h

Example 6

Find the volume of a right cone with a radius of 9\;\mathrm{cm} and a height of 16\;\mathrm{cm}.

Use the formula:

V &= \frac{1}{3} Bh \\ &= \frac{1}{3} \pi r^2 h\\ &= \frac{1}{3} (3.14) (9)(9)(16)\\ &= 1356.48 \ \text{cubic cm}

Example 7

Find the volume of a right cone with a radius of 10\;\mathrm{feet} and a slant height of 13\;\mathrm{feet}.

Use the Pythagorean theorem to find the height:

r^2 + h^2 &= l^2\\ (10)^2 + h^2 &= (13)^2\\ h^2 &= (13)^2 - (10)^2\\ h^2 &= 69\\ h &= 8.31

Now use the volume formula.

V &= \frac{1}{3} \pi r^2 h\\ &= \frac{1}{3} (3.14) (10)(10)(8.31)\\ &= 869.78 \ \text{cubic feet}

Last modified: Tuesday, July 6, 2010, 12:06 PM