Surface Area of a Cylinder — Formula & Examples

Reach for the cylinder surface-area method whenever you need the outside covering of a closed cylindrical object: the metal in a can, the label wrapped around a container, or the paint on a cylindrical part. The method works exactly when the shape is a closed right circular cylinder. If you only need the side, or one base is missing, you adjust a single term at the end.

When this method applies

Use the full method below when the object is a closed right circular cylinder and you want its total outside area. Watch for the variations, because they change which terms you keep:

• Closed cylinder: keep everything.
• Curved side only: use just the lateral term $2\pi rh$.
• One base missing: subtract $\pi r^2$.
• Both bases missing: the answer is only the lateral area.
• Not a right circular cylinder: the formula is just an approximation.

The target formula for a closed cylinder with radius $r$ and height $h$ is

The steps, and where the formula comes from

Step 1 — Identify the radius and height. Use the radius $r$ and height $h$. If the diameter is given, convert first with $r = d/2$.

Step 2 — Account for the two bases. The top and bottom are circles, each of area $\pi r^2$, so together they give

Step 3 — Account for the curved side. Picture the side unrolled as a rectangle wrapped around the cylinder. Its height is $h$ and its width is the base circumference $2\pi r$, so the lateral area is

Step 4 — Add the parts. For a closed cylinder, combine the two circles and the wrapped rectangle:

Step 5 — Check units and condition. Surface area uses square units, and the result changes if the cylinder is open at the top or bottom.

The mental anchor for the whole method: two circles plus one wrapped rectangle.

A full worked example: radius $3$ cm, height $8$ cm

Take a closed cylinder with radius $3$ cm and height $8$ cm and walk every step.

Write the formula:

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

Compute the two parts:

So the exact surface area is $66\pi\ \text{cm}^2$. For a decimal, use $\pi \approx 3.1416$:

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

Where each step tends to break down

If you get stuck, the failure is almost always at one of these checkpoints, so verify them in order:

• Radius step: Using the diameter as if it were the radius. If $d = 6$, then $r = 3$, not $6$.
• Bases step: Dropping the two bases and reporting only $2\pi rh$. That lateral value is a self-check: for a closed cylinder the total must be larger than $2\pi rh = 2\pi(3)(8) = 48\pi$. If your total is not bigger, you skipped the bases.
• Add step: Forgetting the formula changes when the cylinder is open at one or both ends.
• Units step: Writing cubic units. Surface area is square units such as $\text{cm}^2$ or $\text{m}^2$. And do not confuse surface area (outside) with volume (inside).

Your turn

Run a closed cylinder with radius $5$ cm and height $12$ cm through the same five steps: find the side area, then add the two bases, then confirm your total exceeds the lateral value alone.