Faces, edges and vertices

Many solids (cuboid, parallelopiped, cylinder, cone, prism, pyramid) can be made by folding flat material: paper, cardboard, even metal sheet. That is how hollow solids are made in factories.

To talk about them we need three words:

• Faces: the flat surfaces on the outside.
• Edges: the line segments where two faces meet.
• Vertices: the points (corners) where edges meet.

A cube or cuboid has 6 faces, 8 vertices and 12 edges. An easy way to remember: 6, 8, 12, and the biggest number goes with edges. Learn these by heart; everything else builds on them.

A prism has two identical polygons as opposite faces, with edges joining their matching corners. All its other faces are parallelograms. It is named after those polygons: triangular prism, pentagonal prism, hexagonal prism.

A pyramid has one polygon as its base and a single point above it (the apex), with edges from that point to every corner of the base. It is named after its base: triangular pyramid, square pyramid, pentagonal pyramid. A triangular pyramid is also called a tetrahedron, which has 4 faces in all.

Now count for any prism whose polygons have n sides:

• Faces: 2 polygons + one parallelogram per side = n + 2.
• Vertices: n on each polygon = 2n.
• Edges: n round each polygon + n joining them = 3n.

Check with a cube (n = 4): 6 faces, 8 vertices, 12 edges. Correct.

And for a pyramid on an n-sided base:

• Faces: the base + one triangle per side = n + 1.
• Vertices: n on the base + the apex = n + 1.
• Edges: n round the base + n going up to the apex = 2n.

Check a triangular pyramid (n = 3): 4 faces, 4 vertices, 6 edges: the tetrahedron.

Careful: these numbers depend on n. A decagonal prism (n = 10) has 12 faces, 30 edges and 20 vertices, but a hexagonal prism has 8, 18 and 12. Always work from the rule, never from one remembered example.