Volume and Surface Area of Prisms, Cylinders, Cones and Spheres

Q: What is the formula for the volume of a cylinder?

V = π r^2 h: the area of the circular base times the height.

One idea that gives the volume of every prism and cylinder, why pointed solids hold a third as much, the sphere formula, and how to find surface area by unfolding a solid into a net.

3 min read Supports 4 lessons

Area counts unit squares covering a flat shape. Volume counts unit cubes filling a solid. A box $3$ by $3$ by $3$ holds $27$ unit cubes, which is why $3^3$ is called "three cubed".

$3^{3} = 3 \times 3 \times 3 = 27$

$3 \times 3 \times 3 = 27$ cubic units.

Prisms and cylinders

A prism has the same cross-section all the way through, like a stick of rock. Think of it as a stack of identical thin slices. Each slice has the area of the cross-section, and there is a length's worth of them.

Key idea

For any prism or cylinder: volume = cross-sectional area × length.

For a box, the cross-section is a rectangle, giving $lwh$ . For a cylinder, it is a circle, giving $\pi r^2 h$ .

Worked example

Find the volume of a cylinder with radius $3$ and height $10$ .

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Base area.
$\pi \times 3^2 = 9\pi$

Cones and pyramids: one third

A cone fits exactly inside a cylinder with the same base and height, and it holds exactly one third as much. Fill a cone with water three times and you fill the cylinder. The same is true of a pyramid inside a prism.

Spheres

A sphere fits snugly inside a cylinder of the same width and height. Archimedes showed it holds two thirds of that cylinder, which works out as $\frac43\pi r^3$ .

Volumes

\text{box } lwh \qquad \text{cylinder } \pi r^2 h \qquad \text{cone } \tfrac13\pi r^2 h \qquad \text{sphere } \tfrac43\pi r^3

$\text{cylinder: } r = 3,\; h = 6$
Same radius and height for all three.

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Surface area

Surface area is the total area of every face, as if you were wrapping the solid in paper. Unfold the solid flat into a net and add up the areas of the pieces.

A cylinder unrolls into two circles and a rectangle. The rectangle's width is the circumference, $2\pi r$ , so its area is $2\pi rh$ :

\text{cylinder surface area} = 2\pi r^2 + 2\pi r h

Watch out

Volume is in cubic units (cm³); surface area is in square units (cm²). Keep them apart.

Tip

Scaling every length by $k$ multiplies surface area by $k^2$ and volume by $k^3$ . A box twice as big in every direction holds eight times as much.

Common questions

What is the formula for the volume of a cylinder?: $V = \pi r^2 h$ : the area of the circular base times the height.
Why is a cone a third of a cylinder?: A cone with the same base and height as a cylinder holds exactly one third of its volume. Calculus proves it, and you can check it by pouring water.

Practise volume and surface area

The Volume & Surface Area lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the formula for the volume of a cylinder?

$V = \pi r^2 h$ : the area of the circular base times the height.

Why is a cone a third of a cylinder?

A cone with the same base and height as a cylinder holds exactly one third of its volume. Calculus proves it, and you can check it by pouring water.