A solid figure takes up space, so it has length, width and height. Every solid made of flat pieces is described by three counts: faces are the flat surfaces, edges are the lines where two faces meet, and vertices are the points where edges meet.
Slide a square straight upwards and it sweeps out a cube. Slide a circle upwards and it sweeps out a cylinder. Pull every point of a square up to a single point above it and you get a pyramid. Solids are flat shapes with one more instruction added.
The two big families
| Prism | Pyramid | |
|---|---|---|
| How it is made | A flat shape slid straight along | A flat shape pulled up to one point |
| Ends | Two identical ends, parallel to each other | One base and one apex point |
| Side faces | Rectangles | Triangles, all meeting at the apex |
| Cut it anywhere across | Always the same shape and size | Same shape, getting smaller towards the top |
| Named after | Its cross section: triangular prism, hexagonal prism | Its base: square pyramid, triangular pyramid |
Imagine slicing the solid straight across, parallel to its base. If every slice is exactly the same, it is a prism. If the slices get smaller as you move up towards a point, it is a pyramid.
Solids with curved surfaces
A cylinder is a circle slid straight along. A cone is a circle pulled up to a point. A sphere is every point at the same distance from a centre, which is the circle rule from Unit 4 carried into three directions.
Strictly a face means a flat surface, so a cylinder has two faces and one curved surface. Different books count this differently. Whatever your book says, be consistent within one answer and state which you are counting.
One equation ties the three counts together
Count the faces, vertices and edges of any solid with flat faces and no holes, and faces plus vertices always comes to two more than edges. This is Euler's formula:
\[ F + V - E = 2 \]
It is a free check. A cube has six faces, eight vertices, twelve edges: 6 + 8 - 12 = 2. For a solid you find hard to count, count two of the three carefully and let the formula predict the third. If your own count does not give two, you have miscounted, and you know it before the marker does.
It needs flat faces and no holes. A cylinder, cone and sphere have curved surfaces, so the formula does not apply. Neither does it apply to a solid with a tunnel through it, like a nut for a bolt.
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Count the faces, vertices and edges of a triangular prism, and check with Euler's formula.
A hexagonal pyramid: work out all three counts without holding one.
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A solid figure has length, width and height, so it takes up space.
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