Volume of Cylinders

Q: What units does volume use?

Cubic units, like $\text{cm}^3$ or $\text{in}^3$. Volume counts how many unit cubes fit inside the solid, so the units are cubed — surface area, by contrast, uses square units.

A cylinder is a can shape: a circle on the bottom, the same circle on top, and straight sides connecting them. Its volume — how much it holds — comes from one formula: $V = \pi r^2 h$ , where $r$ is the radius of the circular base and $h$ is the height.

The formula makes sense if you read it in two pieces. The base is a circle, and a circle's area is $\pi r^2$ . Stack that circle up to a height of $h$ and you've filled the whole can. So volume is really just base area times height — the same idea as any prism, with a circle for the base.

The formula

Write $V = \pi r^2 h$ at the top of every problem. Then substitute: replace $\pi$ with $3.14$ (unless the problem says otherwise), replace $r$ with the radius, and replace $h$ with the height. The figure below labels the two measurements the formula needs — the radius $r$ of the circular base and the height $h$ .

The exponent belongs to the radius only. Square $r$ before multiplying anything else — in $V = 3.14 \cdot 3^2 \cdot 4$ , compute $3^2 = 9$ first, then multiply straight across. Squaring the whole product instead of just the radius is the most common wrong answer on this topic.

Volume is measured in cubic units: $\text{cm}^3$ , $\text{in}^3$ , $\text{ft}^3$ . If a problem's answer choices carry units, they should be cubed.

The diameter trap

Problems love to give you the diameter instead of the radius. The formula needs the radius — half the diameter. If a can has a diameter of $10$ , its radius is $5$ , and $5$ is what gets squared.

Make it a habit: before substituting, ask whether the number you were given crosses the whole circle (diameter) or goes from the center to the edge (radius). Halve the diameter first, then start the formula.

Worked examples

Example 1: radius and height given

Find the volume of a cylinder with radius $2$ and height $5$ . Use $\pi \approx 3.14$ .

Write the formula

V = \pi r^2 h

Substitute the values

V = 3.14 \cdot 2^2 \cdot 5

Square the radius first

V = 3.14 \cdot 4 \cdot 5

Multiply

V = 62.8

Answer: $V = 62.8$

Example 2: the diameter is given

Find the volume of a cylinder with diameter $10$ and height $3$ . Use $\pi \approx 3.14$ .

Halve the diameter to get the radius

r = \dfrac{10}{2} = 5

Write the formula

V = \pi r^2 h

Substitute the values

V = 3.14 \cdot 5^2 \cdot 3

Square the radius, then multiply

V = 3.14 \cdot 25 \cdot 3 = 235.5

Answer: $V = 235.5$

Example 3: a real container

A soda can is a cylinder with a diameter of $6$ centimeters and a height of $12$ centimeters. How much does it hold? Use $\pi \approx 3.14$ .

Halve the diameter to get the radius

r = \dfrac{6}{2} = 3

Write the formula

V = \pi r^2 h

Substitute the values

V = 3.14 \cdot 3^2 \cdot 12

Square the radius, then multiply

V = 3.14 \cdot 9 \cdot 12 = 339.12

Answer: $V = 339.12 \text{ cm}^3$

Try one yourself

Common questions

Why is the radius squared in the formula?

Because the base of a cylinder is a circle, and a circle's area is $\pi r^2$ . The volume formula is base area times height, so the squaring comes from the circle — the height is never squared.

What if the problem gives the diameter?

Divide it by $2$ to get the radius before you substitute. The formula only works with the radius. A diameter of $8$ means $r = 4$ , and $4$ is what you square.

What units does volume use?

Cubic units, like $\text{cm}^3$ or $\text{in}^3$ . Volume counts how many unit cubes fit inside the solid, so the units are cubed — surface area, by contrast, uses square units.

Should I use 3.14 or the π button on my calculator?

Follow the problem. Most problems at this level say to use $\pi \approx 3.14$ , and the answer choices are computed from $3.14$ . If you use the calculator's $\pi$ button instead, your answer will be slightly off from the choices.

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Volume of Cylinders

The formula

The diameter trap

Worked examples

Example 1: radius and height given

Example 2: the diameter is given

Example 3: a real container

Try one yourself

Common questions

Keep learning

Volume of Cones & Spheres

Surface Area of Prisms & Cylinders

Missing Dimensions & Composite Solids

Area of Circles