You know the area of a circle, $\pi r^{2}$, and the volume of a prism, base area times height. Put them together for a cylinder.
Why it works
A prism’s volume is a stack of copies of its base, one layer on top of another. A cylinder is a stack of circles, like a roll of coins. Each layer has area $\pi r^{2}$, and there are $h$ units of layers, so $V = \pi r^{2} \cdot h$.
Key idea
$$V = \pi r^{2} h$$ where $r$ is the radius of the base and $h$ is the height. Leave $\pi$ in the answer for an exact value, or use $\pi \approx 3.14$ to estimate.
Worked examples
Worked example: Radius and height
Find the volume of a cylinder with radius 3 cm and height 5 cm.
- Base area: $\pi \cdot 3^{2} = 9\pi$.
- Volume: $9\pi \cdot 5 = 45\pi$.
- $45 \times 3.14 = 141.3$, so about 141 cubic centimeters.
Answer: $45\pi \approx 141$ cm³
Worked example: Given the diameter
A can has diameter 10 cm and height 4 cm. Find its volume.
- The radius is half the diameter: $r = 5$.
- $V = \pi \cdot 5^{2} \cdot 4 = 100\pi \approx 314$.
Answer: $100\pi \approx 314$ cm³
Worked example: Work backward
A cylinder has volume $72\pi$ and radius 3. Find its height.
- $72\pi = \pi \cdot 9 \cdot h$.
- Divide by $9\pi$: $h = 8$.
Answer: $8$
Common mistake
Using the diameter in place of the radius. With $d = 10$, using 10 gives $\pi \cdot 100 \cdot 4 = 400\pi$, four times too big, because squaring doubles the error twice. Halve the diameter first.