A geometric solid is a three dimensional shape. The ones met in elementary school divide neatly into those with flat faces, called polyhedra, and those with curved surfaces, which are not.
Faces, edges and vertices
- A face is a flat surface.
- An edge is where two faces meet.
- A vertex is a corner where edges meet. The plural is vertices.
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular prism | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square pyramid | 5 | 8 | 5 |
| Triangular pyramid | 4 | 6 | 4 |
The pattern
For every solid in that table, faces plus vertices minus edges comes to exactly two. Check the cube: 6 + 8 - 12 = 2. Check the triangular prism: 5 + 6 - 9 = 2. The relationship holds for every polyhedron and is a useful way to catch a miscount, since getting a number other than two means one of the three counts is wrong.
The rule applies only to solids with flat faces. A cylinder, a cone and a sphere have curved surfaces, so it does not apply to them and there is nothing to check.
Prisms and pyramids
A prism has two identical parallel bases joined by rectangles, and it is named for the shape of those bases. A pyramid has one base and triangular faces meeting at a single apex, and it is also named for its base. A square pyramid has a square base, and a triangular pyramid has a triangular one.
Curved solids
- Cylinder. Two circular bases joined by a curved surface.
- Cone. One circular base and a curved surface rising to a point.
- Sphere. Every point on its surface the same distance from the centre. No faces, no edges, no vertices.
Common mistakes
Forgetting the hidden parts
A drawing of a cube shows only three faces. All six are still there, and counting only what is visible is the usual reason a count comes out wrong.
Naming a prism after its side faces
A prism is named for its bases, not the rectangles around it.
Check yourself
How many edges does a square pyramid have?
Eight, four around the base and four rising to the apex.
A solid has 6 faces and 12 edges. How many vertices?
Eight, since faces plus vertices minus edges is two.
Which solid has no vertices at all?
A sphere. It has no edges or flat faces either.
What is the difference between a triangular prism and a triangular pyramid?
The prism has two triangular bases joined by three rectangles, five faces in all. The pyramid has one triangular base and three triangular faces meeting at a point, four faces in all.
See also
Sources
- Solid figures, their parts, and the relationship among faces, edges and vertices as taught in United States elementary geometry. Examples on this page are our own.