Solid Shapes, Measured

Everything so far in geometry has been flat. This track steps into three dimensions and measures what it finds there. It opens by naming the parts — faces, edges and vertices — which means learning to count the ones hidden round the back, the single hardest thing about drawing solids on a flat page. The net removes that difficulty entirely: cut a solid along enough edges, press it flat, and every face lies side by side at its true size with nothing foreshortened and nothing hidden. Reading a net backwards is a skill worth having, and it carries one warning that a quick count will never give you — six squares are sometimes a cube and sometimes nothing at all, because the arrangement has to fold without overlapping. From there the track asks the two questions you can ask about any solid, and keeps them firmly apart. Surface area is how much it takes to cover the thing, which is simply the area of its net; for a cuboid the six faces form three matching pairs, so three areas are added and doubled. Volume is how much it holds, counted in unit cubes: one layer, repeated for every layer. The track ends by generalising that layer picture. Any solid with the same cross-section all the way along is a prism, and its volume is that cross-section times its length — a single rule that absorbs the cuboid and the cylinder as special cases.