ShapesFaces, edges and vertices of solid shapes
Solids are described by three counts. A face is a flat or curved surface of the solid. An edge is a line where two faces meet. A vertex (plural vertices) is a corner, a point where edges meet. This is one of the first topics in solid shapes, from class 3 onwards.
Solids with flat faces
| Solid | Faces | Edges | Vertices | F + V − E |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | 2 |
| Cuboid | 6 | 12 | 8 | 2 |
| Triangular prism | 5 | 9 | 6 | 2 |
| Pentagonal prism | 7 | 15 | 10 | 2 |
| Hexagonal prism | 8 | 18 | 12 | 2 |
| Triangular pyramid (tetrahedron) | 4 | 6 | 4 | 2 |
| Square pyramid | 5 | 8 | 5 | 2 |
| Pentagonal pyramid | 6 | 10 | 6 | 2 |
| Hexagonal pyramid | 7 | 12 | 7 | 2 |
| Octahedron | 8 | 12 | 6 | 2 |
Euler's formula
Look at the last column. For every solid with flat faces and straight edges,
Faces + Vertices − Edges = 2
This is Euler's formula, named after the mathematician Leonhard Euler. For a cube: 6 + 8 − 12 = 2. It gives a quick check on your counting, and it answers questions such as: "A solid has 7 faces and 10 vertices. How many edges?" From 7 + 10 − E = 2, there are 15 edges. It is a pentagonal prism.
Solids with curved surfaces
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
| Hemisphere | 2 (1 flat, 1 curved) | 1 | 0 |
Text books differ on how to count these. Most Indian primary books count a curved surface as a face and a circular rim as an edge, which gives the figures above. Euler's formula is meant for solids with flat faces, so do not expect it to hold here.
Prisms and pyramids: the patterns
A prism whose end has n sides has n + 2 faces, 3n edges and 2n vertices. A pyramid whose base has n sides has n + 1 faces, 2n edges and n + 1 vertices. So you never need to memorise each solid: count the sides of the end and use the pattern.
How to count without losing your place
- Faces: count the top and bottom, then go round the sides.
- Vertices: count the corners of the top, then of the bottom.
- Edges: count the edges round the top, then round the bottom, then the upright ones joining them.
- Check with Euler's formula.
Nets
A net is a flat pattern that folds up into a solid. A cube has 11 different nets, each made of six squares. The net of a cylinder is a rectangle with two circles. The net of a cone is a sector with a circle. Cutting open a carton along its edges is the best way to see a net, and to see that the surface area of a solid is simply the area of its net.
Shapes around you
A brick is a cuboid, a die is a cube, a tin is a cylinder, a ball is a sphere, a birthday cap is a cone, and a tent is often a triangular prism. Asking a child to name the solid and count its faces, edges and corners turns any room into a lesson.
For areas and volumes, see surface area and volume, or open a solid from the home page.