Solid nets & surface area

A cardboard box can be folded flat, and that is exactly where the whole trick lies. Cut an empty cereal box open along a few edges and unfold it: what remains is a flat figure made of rectangles. From that flat figure you can read off how much cardboard the box uses, without having to wrestle with space. That is what this topic is about: from the solid to its net, and from the net to the surface area.

From solid to net

A net is the solid unfolded: every face laid out side by side in the plane, still connected so that you can fold it back into the solid. For a cube that means 6 equally large squares, because a cube has a face on top and bottom, front and back, left and right. A cuboid also has 6 faces, but now they are rectangles, and they come in three pairs: opposite faces are always the same size. Counting is your first check. If your net does not have 6 parts, a face is missing or you have drawn one twice.

Surface area means: add up every face

The surface area $O$ is nothing more than the sum of all the faces in the net. For a cube with edge length $a$, all 6 faces are squares of area $a^2$, so $O = 6 \cdot a^2$. For $a = 4$ cm you get $6 \cdot 16 = 96$, so the surface area is 96 cm². For a cuboid with edges $a$, $b$ and $c$ there are three different rectangles, and each one appears twice: $a \cdot b$, $a \cdot c$ and $b \cdot c$. A box measuring 5 cm, 3 cm and 2 cm therefore has surface area $2 \cdot (15 + 10 + 6) = 62$, that is 62 cm². This also works backwards: if you know the surface area of a cube, you divide it by 6 and look for the side length of the square with exactly that area.

The cylinder unrolls

A tin can is made of a lid, a base and the wall in between. If you cut the wall open along a vertical line and unroll it, you get a rectangle. Its height is the height $h$ of the can, and its width is exactly the circumference $2 \pi r$, because that is how far the wall reaches all the way round. On top of that come the lid and the base, two circles of $\pi r^2$ each. For $r = 3$ cm and $h = 10$ cm the rectangle is $2 \cdot 3.14 \cdot 3 = 18.84$ cm wide and 10 cm high, so 188.4 cm². The two circles add up to $2 \cdot 3.14 \cdot 9 = 56.52$ cm², giving 244.92 cm² in total (using $\pi \approx 3.14$).

Surface area or volume?

Surface area and volume answer two completely different questions. The surface area tells you how much material the shell needs: wrapping paper, paint, sheet metal. It is made of faces, so it is given in cm² or m². The volume tells you how much fits inside: water, cereal, air. It counts little cubes and is given in cm³ or litres. A cube with 4 cm edges has a surface area of 96 cm² and a volume of 64 cm³. Same solid, two completely different numbers with different units.