Volume and surface area of a cylinder

A cylinder is the tin-can shape: a circle at the bottom, a circle at the top, and a curved surface between them. The volume can be found if we have the height and the radius, and the surface area needs the same two numbers.

When do I use this?

Whenever a task asks how much fits inside a cylinder, or how much material covers one. Volume is inside, surface area is outside. The general idea is explained in 3D shapes: volume and surface area.

The volume of a cylinder

The volume of a cylinder can be found if we have the height and the radius. We use this formula:

$$V = \pi \times r^2 \times h$$

What we're actually doing is finding the area of the circle at the bottom, $\pi \times r^2$, and then multiplying by how tall the cylinder is. It is the same move as for the cuboid: the area of the bottom times the height.

Take a cylinder with radius $3$ cm and height $10$ cm. Then

$$V = \pi \times 3^2 \times 10 = \pi \times 9 \times 10 = 90\pi \approx 282.74 \text{ cm}^3$$

So about $282.74$ cubic centimetres fit inside it. If the question asks for the answer in terms of $\pi$, we stop at $90\pi$.

The surface area of a cylinder

To find the surface area we use the formula:

$$S = 2 \times \pi \times r \times h + 2 \times \pi \times r^2$$

Here we first find the area of the curved surface, the tube. We can see the tube as a rectangle that has been folded round into a ring, where one side is the height, and the other side is the circumference of the circle, $2 \times \pi \times r$. That gives the first part, $2 \times \pi \times r \times h$.

Then we add the area of the circle at the bottom and the circle at the top. Each is $\pi \times r^2$, so together they are $2 \times \pi \times r^2$. That is the second part.

For the same cylinder, with $r = 3$ and $h = 10$:

$$S = 2 \times \pi \times 3 \times 10 + 2 \times \pi \times 3^2 = 60\pi + 18\pi = 78\pi \approx 245.04 \text{ cm}^2$$

So about $245.04$ square centimetres of wrapping paper would cover the cylinder.

The two answers side by side

For the cylinder with radius $3$ cm and height $10$ cm:

• Volume: $V = 90\pi \approx 282.74$ cm³.
• Surface area: $S = 78\pi \approx 245.04$ cm².

Two different questions, two different units. The volume is cubic because it is inside; the surface area is square because it is an area.

Common mistakes

• Mixing up volume and surface area. Volume is the circle at the bottom times the height. Surface area is the tube plus the two circles. Check which one the question asks for.
• Forgetting the two circles. The tube on its own is $2 \times \pi \times r \times h$. A closed cylinder also has a circle at the bottom and one at the top, and they need adding on.
• Giving the answer in the wrong unit. Volume in cm³, surface area in cm². If the unit is wrong, we have probably calculated wrong somewhere.

Related

All the 3D shapes together are in 3D shapes: volume and surface area. The circle at the bottom is explained in how to find the area of a circle, and the circumference that makes the tube in how to find the circumference of a circle. The cuboid uses the same area-of-the-bottom-times-height idea, in volume and surface area of a cuboid. The cone is the cylinder's pointed cousin, in volume and surface area of a cone.