Shapes that tile the plane
Which shapes fit round every point of a tiling?
A floor of square tiles
A floor can be covered with square tiles and it goes on in every direction. Look at one corner where tiles meet. Four squares meet there.
Why do four squares fit exactly?
What this lesson covers
The idea
Copies of squares, of equilateral triangles, or of regular hexagons can tile the entire plane; a plane can also be tiled using more than one shape and using non-regular polygons.
A floor of square tiles
A floor can be covered with square tiles and it goes on in every direction. Look at one corner where tiles meet. Four squares meet there.
Why do four squares fit exactly?
- 4 × 90° = 360°
- 4 squares have 4 sides
- Squares are the smallest shape
Try the shapes
Pick a shape. The copies at the point O are outlined. Add up their angles. If the total is 360°, the pattern spreads.
360° at every corner
A shape tiles the plane when copies fit round a point with no gap and no overlap, so the angles must add up to 360°. Regular triangles, squares and hexagons do this. Regular pentagons and octagons do not. Two different shapes can also work together: two octagons and a square.
Copies of squares, equilateral triangles or regular hexagons can tile the entire plane. Round each point the angles add up to 360°. A plane can also be tiled using more than one shape.
- Shape | Angle | Round a point
- Triangle | 60° | 6 × 60° = 360°
- Square | 90° | 4 × 90° = 360°
- Hexagon | 120° | 3 × 120° = 360°
- Pentagon | 108° | 3 × 108° = 324°, gap of 36°
- Octagon | 135° | 2 × 135° = 270°, gap of 90°
- Octagon, octagon, square | 135°, 135°, 90° | 360°
Notes
Copies of squares, equilateral triangles or regular hexagons can tile the entire plane, because the angles round a point add up to 360°. A plane can also be tiled using more than one shape.
Check yourself
Each angle of a regular hexagon is 120°. How many hexagons meet at each point of a honeycomb?
Answer: 3
360° ÷ 120° = 3 hexagons.
Which of these regular shapes can not tile the plane by itself?
Two regular octagons meet at a point. Each angle of an octagon is 135°. How many degrees of gap are left round the point?
Answer: 90
2 × 135° = 270° and 360° − 270° = 90°. A square has a 90° angle and fits the gap.
- Equilateral triangle. Six triangles meet at a point: 6 × 60° = 360°.
- Square. Four squares meet at a point: 4 × 90° = 360°.
- Regular pentagon — correct. Yes! Each angle is 108°. 3 × 108° = 324° leaves a gap of 36°, and 4 × 108° = 432° is too much.
- Regular hexagon. Three hexagons meet at a point: 3 × 120° = 360°.