The volume of a cylinder and cone are closely related when they share the same base radius and height. The volume of a cone is exactly one-third the volume of a cylinder with the same dimensions. This relationship makes it easy to calculate one if you know the other! Understanding this connection simplifies solving many geometry problems.
Specifically, if a cylinder has a radius $r$ and height $h$, its volume $V_{cylinder}$ is given by $V_{cylinder} = \pi r^2 h$. A cone with the same radius $r$ and height $h$ has a volume $V_{cone}$ given by $V_{cone} = \frac{1}{3} \pi r^2 h$. Therefore, $V_{cone} = \frac{1}{3} V_{cylinder}$.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Cylinder | A. A three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. |
| 2. Cone | B. The perpendicular distance from the base to the furthest point of a shape. |
| 3. Volume | C. A three-dimensional geometric shape with straight parallel sides and a circular or oval cross-section. |
| 4. Radius | D. The amount of space that a substance or object occupies. |
| 5. Height | E. The distance from the center of a circle to any point on its circumference. |
(Match the numbers 1-5 to the letters A-E)
Complete the following paragraph using the words: one-third, cylinder, cone, volume, radius, height.
The volume of a ______ and a ______ are related when they have the same ______ and ______. The ______ of the cone is ______ the volume of the cylinder.
Explain in your own words why the volume of a cone is one-third the volume of a cylinder with the same base and height. Can you think of a real-world example where this relationship is useful?
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