Surface Area of a Cylinder — Formula & Examples

A cylinder's total surface area is the area of its two circular ends plus its curved side. For a closed right circular cylinder with radius $r$ and height $h$, the formula is

$$A = 2\pi r^2 + 2\pi rh$$

Use this when the cylinder is closed. If the problem asks for only the curved surface, use $2\pi rh$. If the top or bottom is missing, subtract the area of the missing circle.

Surface area of a cylinder formula explained

The formula has two parts because the shape has two different kinds of surfaces.

The top and bottom are circles. Each one has area $\pi r^2$, so together they give

$$2\pi r^2$$

The side is curved, but you can picture it as a rectangle wrapped around the cylinder. Its height is $h$, and its width is the circumference of the base, $2\pi r$. That makes the lateral area

$$(2\pi r)(h) = 2\pi rh$$

Add the circles and the side:

$$A = 2\pi r^2 + 2\pi rh$$

That is the key idea to remember: two circles plus one wrapped rectangle.

Worked example: radius $3$ cm, height $8$ cm

Suppose a closed cylinder has radius $3$ cm and height $8$ cm.

Write the formula:

$$A = 2\pi r^2 + 2\pi rh$$

Substitute $r = 3$ and $h = 8$:

$$A = 2\pi(3^2) + 2\pi(3)(8)$$

Compute the two parts:

$$A = 2\pi(9) + 48\pi = 18\pi + 48\pi$$ $$A = 66\pi$$

So the exact surface area is $66\pi\ \text{cm}^2$.

If you need a decimal approximation, use $\pi \approx 3.1416$:

$$66\pi \approx 207.3\ \text{cm}^2$$

The answer is in square centimeters because surface area measures covering, not space inside.

One quick check that catches a common mistake

If you calculate only the side,

$$2\pi rh = 2\pi(3)(8) = 48\pi$$

you have found the lateral surface area, not the total surface area.

For a closed cylinder, the total must be larger because it also includes two circular bases. This quick comparison is an easy way to catch a setup error before you finish.

Common mistakes with cylinder surface area

1. Using the diameter as if it were the radius. If $d = 6$, then $r = 3$, not $6$.
2. Using only $2\pi rh$ when the question asks for total surface area.
3. Writing cubic units. Surface area should use square units such as $\text{cm}^2$ or $\text{m}^2$.
4. Forgetting that the formula changes if the cylinder is open at the top or bottom.
5. Mixing surface area with volume. Surface area measures the outside; volume measures the space inside.

When to use the surface area formula

Use this formula when you need the outside covering of a closed cylindrical object. Typical examples are the metal needed for a can, the label area around a container, or the painted area on a cylindrical part.

The condition matters. If you need only the side covering, use $2\pi rh$. If one base is missing, subtract $\pi r^2$. If both are missing, the result is just the lateral area. If the shape is not a right circular cylinder, this formula is only an approximation.

Try your own version

Try your own version with radius $5$ cm and height $12$ cm. First find the side area, then add the two circular bases. If you want another next step, solve a similar problem and compare your setup before simplifying.