Surface Area of a Cylinder

Unrolling the side of a cylinder gives a rectangle. The lateral area is 2 × π × radius × height, and the total surface area adds the two circular ends: 2 × π × radius × (radius + height).

Explanation

The surface of a cylinder is made of a curved side and two circular ends. Cut the side open and lay it flat and you get a rectangle whose height is $h$ and whose width is the circumference of the base, $2\pi r$ . That rectangle is the lateral area.

A = 2 \pi r h \qquad S = 2 \pi r (r + h)

$A$ — the lateral area, the curved side on its own
$S$ — the total surface area, side plus both ends
$r$ — the radius of the base
$h$ — the height

The total adds the two circular ends, $2\pi r^2$ , to the lateral area: $2\pi r h + 2\pi r^2 = 2\pi r(r + h)$ .

Example

With the defaults, $r = 3$ and $h = 7$ :

lateral area $A = 2\pi \times 3 \times 7 = 42\pi \approx 131.9469$
total surface area $S = 2\pi \times 3 \times (3 + 7) = 60\pi \approx 188.4956$

As a check, each end has area $\pi \times 3^2 = 9\pi$ , so the two together are $18\pi$ . Adding that to the lateral $42\pi$ gives $60\pi$ , which matches the total.

Watch out

The lateral area is the curved side alone; the total surface area includes both flat ends. The label wrapped around a tin uses the lateral area, while painting the whole tin uses the total. A container with no lid needs one end, not two. Both the radius and the height must be positive.