Faces, edges and vertices

A solid is made of three kinds of part. The faces are its flat surfaces, the edges are the lines where two faces meet, and the vertices are the corners where edges meet. Counting them is how you describe a solid, and the trick is to count in an order, so that nothing is missed or counted twice.

How to count them

1. Choose one kind of part to count, faces, edges or vertices, and count only that.
2. Faces: count the top, then the bottom, then go round the sides. On a drawing, the faces at the back are hidden, so work them out: a box has as many faces at the back as you can see at the front.
3. Edges: count the edges round the top, then round the bottom, then the ones going up the sides. On a drawing, the dashed lines are edges at the back. They count.
4. Vertices: count the corners round the top, then round the bottom. A pyramid has the corners of its base and one more at its point.
5. Older children can check the count with the rule in "Why it always comes out at 2" below.

Worked examples

All the counts in one place

Cube: 6 faces, 12 edges, 8 vertices.

Cuboid: 6 faces, 12 edges, 8 vertices.

Square-based pyramid: 5 faces, 8 edges, 5 vertices.

Triangular-based pyramid: 4 faces, 6 edges, 4 vertices.

Triangular prism: 5 faces, 9 edges, 6 vertices.

Hexagonal prism: 8 faces, 18 edges, 12 vertices.

Corners or vertices

Vertex is the maths word for a corner, and vertices is the word for more than one. Younger children say corners, and the app does too. Both are right.

Why it always comes out at 2

For a solid with flat faces only, there is a check that always works. Add the faces and the vertices, then take away the edges. The answer is always 2. A cube: 6 + 8 − 12 = 2. A square-based pyramid: 5 + 5 − 8 = 2. A triangular prism: 5 + 6 − 9 = 2. It is called Euler's rule, after the mathematician who wrote it down, and it is written F + V − E = 2.

You can see why with a prism. If its end has some number of sides, it has that many rectangles, plus the 2 ends. It has that many edges on each end, plus that many joining the ends, so three lots. And it has that many vertices on each end, so two lots. Faces plus vertices make three lots and 2 more, and taking the three lots of edges away leaves 2. A hexagonal prism: 8 + 12 − 18 = 2.

Cut a corner off a cube and the rule still holds. The cut makes 1 new face and 3 new edges, and the 1 old vertex becomes 3 new ones, so 2 more vertices. 1 + 2 − 3 = 0, so the answer does not change: 7 + 10 − 15 = 2.

If your count does not come out at 2, one of the three numbers is wrong. Usually it is the edges, because the ones at the back are easy to miss.

The rule is only for solids with flat faces. A cylinder or a cone has curved surfaces, and schools count those in different ways, so do not use it on them.

Where mistakes happen

• Forgetting the faces, edges and vertices at the back. A drawing of a cube shows only 3 faces, 9 edges and 7 vertices. The other 3 faces, 3 edges and 1 vertex are hidden behind it.
• Counting an edge twice. Each edge is shared by two faces, so going round each face and adding up its edges gives double. Counting top, bottom, then sides avoids this.
• Mixing up edges and vertices. An edge is a line, and a vertex is a point. A cube has 12 edges but only 8 vertices.
• Calling a face a side. Sides belong to flat shapes. On a solid, the flat surfaces are faces and the lines are edges.
• Forgetting the point of a pyramid. It is a vertex, so a square-based pyramid has 5 vertices, not 4.

Questions people ask

Related

• 3D shapes and their names
• How to count sides and corners
• The names of 2D shapes