Prisms and Pyramids: Faces, Edges, Vertices

There are two families. A prism has two identical ends joined by rectangles, and slicing it anywhere along its length gives the same shape every time. A pyramid has one base and triangles that meet at a point, so slicing it gives a shape that shrinks as you go up. Both are named after that base — a triangular prism, a hexagonal pyramid — and a cube is simply a square prism. Cones and cylinders belong to neither family, because their surfaces curve.

Count a pyramid in groups. A hexagonal pyramid has six triangles round the sides plus the base, seven faces. Its edges come in two groups, six round the base and six running up to the point, twelve altogether. Its vertices are the six base corners plus the apex, seven. The pattern for any pyramid on an n-sided base is n plus one faces, two n edges, and n plus one vertices.

A prism counts the same way, in groups. A hexagonal prism has six rectangles round the sides plus two hexagonal ends, eight faces. Its edges come in three groups of six — round the front, round the back, and the six joining them — eighteen in all. Its vertices are six at each end, twelve. So for an n-sided end: n plus two faces, three n edges, two n vertices. A cube fits the pattern with six faces, twelve edges and eight vertices.

When two solids are glued face to face, both of the glued faces vanish inside the new shape. So add the two counts and take away two. A cube has six faces and a square pyramid five, eleven altogether, so gluing the pyramid onto the top of the cube gives a solid with nine faces. Two cubes glued together have ten, and two square pyramids joined base to base have eight.

You do not need to memorise a table, because the two rules rebuild any row of it in seconds: a pyramid on n sides has n plus one faces, two n edges and n plus one vertices, and a prism on n sides has n plus two faces, three n edges and two n vertices. Two things are worth noticing as checks — a pyramid always has as many vertices as faces, and a prism always has an even number of vertices. Quiz and exercises now.