Properties of 3D Solids

Three special words

Every solid (3D) shape can be described using three words:

• A face is a surface of the solid. It can be flat (like the side of a box) or curved (like the side of a ball).
• An edge is the line where two faces meet.
• A vertex is a corner — a point where edges meet. The plural is vertices.

Described diagram: picture a cube, like a dice. Each flat square you can see is a face. Run your finger along where two squares meet — that line is an edge. The sharp corner points are the vertices.

The cube and the cuboid

A cube has:

• 6 faces (all squares)
• 12 edges (all equal length)
• 8 vertices

A cuboid (a box shape, like a cereal packet) has the same counts — 6 faces, 12 edges, 8 vertices — but its faces are rectangles, so they are not all the same size.

Prisms

A prism has two identical ends joined by flat sides, so if you sliced through it the cross-section would look the same all the way through.

Prism	Faces	Edges	Vertices
Cube / cuboid	6	12	8
Triangular prism	5	9	6

A triangular prism (the shape of a toblerone box) has 2 triangular ends and 3 rectangular faces.

Pyramids

A pyramid has one base and triangular sides that rise to meet at a single point called the apex.

• A square-based pyramid has 5 faces (1 square + 4 triangles), 8 edges and 5 vertices.
• A triangular-based pyramid (a tetrahedron) has 4 faces, 6 edges and 4 vertices.

The quick difference: a prism keeps the same shape through, while a pyramid narrows to a point.

Curved solids

Some solids have curved surfaces, so we describe them a little differently:

• A cylinder (a tin can): 2 flat circular faces and 1 curved surface; 2 curved edges; no vertices.
• A cone (an ice-cream cone): 1 flat circular face and 1 curved surface meeting at an apex; 1 curved edge; 1 vertex (the point).
• A sphere (a ball): 1 curved surface; no flat faces, no edges, no vertices.

Euler's rule: a hidden pattern

For any solid with only flat faces (a polyhedron), there is a beautiful pattern discovered by the mathematician Euler:

Faces + Vertices − Edges = 2

Check it on a cube: 6 + 8 − 12 = 2. ✓ Check it on a triangular prism: 5 + 6 − 9 = 2. ✓

You can even use it to find a missing count. If a solid has 6 faces and 8 vertices, then 6 + 8 − edges = 2, so it has 12 edges.

Activity: build and count

Make solids from straws (edges) and modelling clay balls (vertices). Build a cube, a triangular prism and a square-based pyramid. Count the faces, edges and vertices, then test Euler's rule on each one. You will find it works every time for flat-faced shapes.

Where this connects

This builds on the everyday shapes in 2D and 3D shapes. When you unfold a solid flat, you get its net, which is the next step toward finding its surface area.