Adding up the faces

The surface area of a solid is the total area of all its faces — the amount of wrapping paper it would take to cover it. The easiest way to see every face is to unfold the solid into its net (all the flat pieces laid out), then add up their areas.

A cuboid with length l, width w and height h has 6 rectangular faces in 3 matching pairs: front and back (l\times h), left and right (w\times h), top and bottom (l\times w). Add one of each pair and double:

A cube is the special case where every edge is the same length s, so all 6 faces are identical squares of area s^2:

Rolling out a cylinder

A cylinder unrolls into three flat pieces: the top and bottom are two circles, each of area \pi r^2, and the curved side opens out into a rectangle. That rectangle's height is the cylinder's height h, and its width is the distance once around the circle — the circumference 2\pi r — so its area is 2\pi r \times h:

See the net

Step through the figure. First the cuboid flattens into its six rectangles; then the cylinder unrolls into a rectangle (width 2\pi r) between its two circular ends. The surface area is just the total area of all of these flat pieces.