What is a 3D shape?
A flat shape like a square has only length and width. It is 2D. A 3D shape, or solid, also has height. It takes up space.
- Face: a surface of the solid. It can be flat or curved.
- Edge: a line where two faces meet.
- Vertex (plural vertices): a corner, where edges meet.
A solid with only flat faces, each one a polygon, is called a polyhedron.
Families of solids
| Solid | Faces | Vertices | Edges |
|---|---|---|---|
| Cube | 6 squares | 8 | 12 |
| Cuboid | 6 rectangles | 8 | 12 |
| Triangular prism | 5 | 6 | 9 |
| Square pyramid | 5 | 5 | 8 |
| Triangular pyramid (tetrahedron) | 4 | 4 | 6 |
| Cylinder | 2 flat + 1 curved | 0 | 2 (curved) |
| Cone | 1 flat + 1 curved | 1 (apex) | 1 (curved) |
| Sphere | 1 curved | 0 | 0 |
A prism has two equal, parallel ends (its cross-section stays the same) joined by rectangles. A pyramid has one base and triangular faces that meet at one point, the apex. A prism or pyramid is named after its base.
For a prism with an n-sided base: F = n + 2, V = 2n, E = 3n. For a pyramid with an n-sided base: F = n + 1, V = n + 1, E = 2n.
Nets: solids laid flat
A net is a flat pattern that can be folded to make a solid. Every face appears once in the net.
- A cube has 11 different nets. A row of 4 squares with one square above and one below is the most common.
- A cylinder's net is two circles and one rectangle. The rectangle's length equals the circle's circumference.
- A cone's net is a circle and a sector (a slice of a bigger circle).
- A square pyramid's net is a square with four triangles.
Not every pattern of 6 squares is a cube net. If two squares land on the same side when folded, it fails.
Views and sketches
Engineers draw a solid as three flat pictures: the front view (elevation), the top view (plan) and the side view. For example, a cylinder standing up looks like a rectangle from the front and a circle from the top. A square pyramid looks like a triangle from the front and a square with its diagonals from the top.
On paper we can sketch solids on isometric dot paper, where hidden edges are shown as dashed lines.
Try it at home
Take an empty toothpaste or tea box. Count its faces, edges and vertices. Then carefully cut along some edges so it lies flat. You have made a net. Look at the box from the top, front and side and draw the three views.
Key formulas and definitions
- Euler's rule for polyhedra: F + V − E = 2
- Prism with n-sided base: F = n + 2, V = 2n, E = 3n
- Pyramid with n-sided base: F = n + 1, V = n + 1, E = 2n
- Cube: 6 faces, 8 vertices, 12 edges
- Views: front (elevation), top (plan), side
Worked examples
1. A hexagonal prism. Find F, V and E, and check Euler's rule.
Step 1: n = 6. Step 2: F = 6 + 2 = 8. Step 3: V = 2 × 6 = 12. Step 4: E = 3 × 6 = 18. Check: 8 + 12 − 18 = 2. Correct.
2. A polyhedron has 7 faces and 10 vertices. How many edges?
Step 1: F + V − E = 2. Step 2: 7 + 10 − E = 2. Step 3: 17 − E = 2, so E = 15.
3. A cylinder net has circles of radius 7 cm. How long must the rectangle be?
The rectangle wraps around the circle, so its length = circumference = 2 × π × r = 2 × 22/7 × 7 = 44 cm.
Common mistakes
- Calling a cube's corners 'edges'. Edges are lines; vertices are corner points.
- Thinking any 6 joined squares make a cube net. Fold it in your head to check that no two squares overlap.
- Using Euler's rule for a cylinder, cone or sphere. It is meant for polyhedra (flat faces only).
- Mixing up prisms and pyramids. A prism has two equal ends; a pyramid has one base and a point.