Surface Area of Solids

Adding up the faces

The surface area of a solid is the total area of all of its faces — picture how much wrapping paper it would take to cover the whole thing, with no gaps and no overlaps. The cleverest way to see every face at once is to unfold the solid into its net: cut along some edges and flatten it out, so all the faces lie side by side. The surface area is simply the area of the net.

Because the net is just a collection of flat shapes, you already know how to do this — work out the area of each piece and add them up. Nothing new, just lots of little areas joined together.

a boxed present

Wrapping a present is a surface-area question in disguise. The paper has to cover every face of the box — the top, the bottom and all four sides — so the paper you need is exactly the box's surface area (plus a little to overlap and tuck in). Forget the bottom face and the present slides straight out the bottom!

Cuboids and cubes

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). Each pair is identical, so work out one of each and double the lot:

\text{SA} = 2(lw + lh + wh)

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:

\text{SA} = 6s^2

Worked examples

A cube. A dice has side s = 2\text{ cm}. Each of its 6 faces is a 2\times 2 square of area 4\text{ cm}^2, so

\text{SA} = 6 \times 2^2 = 6 \times 4 = 24\text{ cm}^2.

A cuboid. A box is 5\text{ cm} long, 3\text{ cm} wide and 2\text{ cm} high. The three pairs are top & bottom 5\times 3 = 15, front & back 5\times 2 = 10, and the two sides 3\times 2 = 6:

\text{SA} = 2(15 + 10 + 6) = 2 \times 31 = 62\text{ cm}^2.

A bigger cube. A storage crate has side 10\text{ cm}:

\text{SA} = 6 \times 10^2 = 6 \times 100 = 600\text{ cm}^2.

The two traps that catch everyone:

Add up every face. It is easy to forget the back, the bottom and the far side, because you can't see them in a drawing — but the paper still has to cover them. A cuboid always has 6 faces, and so does a cube.
Surface area is an area, so it is measured in squares — \text{cm}^2. That is different from the volume, the space inside, which is measured in cubes — \text{cm}^3. Same box, two completely different measurements.

a bus a train carriage

A painter pricing a job needs the surface area, not the volume. To paint a big wooden crate — or the outside of a bus or a train carriage — you cover all six faces, so you work out 2(lw + lh + wh) and buy enough paint for that many square centimetres. The empty space inside doesn't cost a single drop of paint.

Rolling out a cylinder

The same "unfold and add" idea works for a cylinder, which opens 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:

\text{SA} = 2\pi r^2 + 2\pi r h

Surface area is always the total area of all the faces — unfold the net and add them up:

cuboid (l \times w \times h): \text{SA} = 2(lw + lh + wh);
cube (side s): \text{SA} = 6s^2;
closed cylinder (radius r, height h): \text{SA} = 2\pi r^2 + 2\pi r h.

See it: the net, face by face

Here is a cuboid unfolded into its six rectangles, with the area written on every face. The two ends match, the front and back match, the top and bottom match — and the total at the bottom is the surface area. Press Refresh for a new box (sometimes a cube, where all six faces are equal).

See it unfold

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.

See it in 3-D

The surface area of a solid is the total area of all its faces. A cuboid has three pairs of matching rectangles — here each pair is shown in a different tint, so the pairs of equal faces are easy to tell apart. Add up all six and you have the surface area. Drag to rotate the box and find every face — including the back, the bottom and the far side you can't see in a flat drawing.