A cone has a circle for a base, like a cylinder, but it narrows to a point. How much less does it hold?
Why it works
Fill a cone with water and pour it into a cylinder with the same base and the same height. It takes exactly three cones to fill the cylinder. So a cone holds one third of its matching cylinder. (The same is true for a pyramid and its prism; a careful proof comes in calculus.)
Key idea
$$V = \frac{1}{3}\pi r^{2} h$$ where $h$ is the height measured straight up from the center of the base, not the slant height along the side.
Worked examples
Worked example: Radius and height
A cone has radius 3 and height 4. Find its volume.
- Matching cylinder: $\pi \cdot 3^{2} \cdot 4 = 36\pi$.
- One third of it: $12\pi$.
Answer: $12\pi$
Worked example: Given the slant height
A cone has radius 6 cm and slant height 10 cm. Find its volume.
- Height, radius and slant height form a right triangle: $h^{2} + 6^{2} = 10^{2}$, so $h = 8$.
- $V = \frac{1}{3}\pi \cdot 36 \cdot 8 = 96\pi \approx 301$.
Answer: $96\pi \approx 301$ cm³
Worked example: Compare with a cylinder
A cylinder holds $90\pi$ cubic inches. How much does a cone with the same base and height hold?
- A cone is one third of the matching cylinder: $90\pi \div 3$.
Answer: $30\pi$ cubic inches
Common mistake
Forgetting the $\frac{1}{3}$, which gives the volume of the whole cylinder, or using the slant height as $h$. The height must be perpendicular to the base; if you are given the slant height, use the Pythagorean theorem to find $h$ first.