Surface Area of a Cylinder

Two circles and a curved side

A cylinder has three surfaces: a circular top, a circular bottom and a curved side joining them. Think of a tin of soup. The top and the bottom are metal circles, and the curved side is covered by the paper label wrapped round it.

The surface area of the cylinder is the total area of all three. Each end is a circle of radius r, so each has area $\pi r^2$πr². The curved side needs more thought, because it is not flat.

A cylinder: two circular ends and one curved side.

Peel off the label

Cut the label straight down, from the top rim to the bottom rim, and peel it off. It lies flat as a rectangle, and peeling it off does not stretch it, so its area has not changed.

One side of the rectangle is the height of the cylinder, h. The other side went once round the rim of the tin, so its length is the circumference of the circle, $2\pi r$2πr. The area of the rectangle is $2\pi r\times h$2πr × h, so the curved surface area of a cylinder is $2\pi$2πrh.

The label laid flat: a rectangle h tall and $2\pi r$2πr wide, the circumference of the tin.

Drag the lower handle to peel off the lids and unroll the curved side. Drag the upper one to change the radius, and watch the width of the rectangle change with it.

Add the two ends

Put the two circles back with the rectangle, and the whole surface is covered: that is the net of the cylinder. The total surface area of a cylinder is $2\pi r^2+2\pi$2πr² + 2πrh, which can also be written $2\pi r\left(r+h\right)$2πr(r + h).

A cylinder with a radius of 3 cm and a height of 5 cm has two ends of $\pi \times 3^2=9\pi$π × 3² = 9π cm² each, which is $18\pi$18π cm², and a curved surface of $2\pi \times 3\times 5=30\pi$2π × 3 × 5 = 30π cm². Its surface area is $18\pi +30\pi =48\pi$18π + 30π = 48π cm², which is about 150.8 cm².

Read the question for which surfaces it wants. A tin with a lid has both ends. An open tin, or a glass, has only one end: $\pi r^2+2\pi$πr² + 2πrh. A pipe open at both ends has only the curved surface, $2\pi$2πrh.

The net of a cylinder: two circles and a rectangle.

Two slips

Making the rectangle as wide as the diameter, 2r, instead of the circumference, $2\pi r$2πr. The label wraps all the way round the tin, and the distance round is $\pi$π times the distance across.

Forgetting the ends. For the cylinder above, the curved surface alone is $30\pi$30π cm², and the two ends add another $18\pi$18π cm².