Nets and surface area

A cube with 4 cm edges is wrapped by exactly 96 cm² of card, and you can see all of it at once when the cube is opened flat.

What you see when a delivery box is pressed flat

Before a cardboard box goes out with the recycling, most of us slit the tape and press it flat. Look at what is lying on the floor. Every face of the box is still there, joined to its neighbours along the edges you did not cut. That flat shape is called a net. Its total area is the surface area of the box: the amount of card the factory needed, if you ignore the flaps glued on the inside.

Surface area wraps a solid, but it is still an area, so it is measured in square units such as cm² or m². Volume asks something else. How much space is inside? Its units are cubic, such as cm³. The two do not have to grow together, either. A pizza box is wide and thin, so it has a large surface area and a small volume, and a chunky box can be the other way round. That is why this course puts both formulas for each family of solids side by side; they are easier to remember in pairs.

Six equal squares add up to 96 cm²

The cube is the simplest case. It has six square faces, all the same size. If each edge is a, one face is a × a = a², and the whole cube needs 6a² of card. The cube in the model has 4 cm edges, so it needs 6 × 16 = 96 cm². Press Unfold and count. Four squares run around the sides in a row, one sits on top and one underneath. Nothing overlaps. Nothing is missing. Those two checks are exactly what make a flat shape a net.

Why six squares in a row will not fold into a cube

Cut a different set of edges and you still get one flat piece, so a cube has more than one net. Counted carefully, with turned or flipped copies treated as the same, there are 11 of them. Six squares in one straight row is not among them. Fold it and the first and last squares land on the same side, which leaves a tube with two open ends. To test a shape, fold it in your head. Each square needs its own place on the cube, and two faces that sit opposite each other on the cube can never share an edge in the net.

A cuboid has three pairs of matching rectangles

Most delivery boxes are not cubes. A rectangular prism, or cuboid, has length l, width w and height h. Its six faces come in three matching pairs: top and bottom are l × w, front and back are l × h, and the two ends are w × h. Take one face from each pair, add them, and double the total: surface area = 2(lw + lh + wh). A box 6 cm long, 4 cm wide and 3 cm high needs 2(24 + 18 + 12) = 108 cm². Here is a made-up parcel box, 40 cm × 30 cm × 20 cm. It needs 2(1,200 + 800 + 600) = 5,200 cm² of card, a little over half a square metre. Switch the model to the rectangular prism and open it, and the three pairs are easy to spot.

Packaging factories print nets on big flat sheets, cut them out with a die and fold them by machine. Two nets of the same box always have the same total area. Their outlines differ, though, and one outline may nest against its neighbours on the sheet with less waste than another. A designer who knows several nets of one box simply has more ways to save card.