geometry · Year 6 · AC9M6SP03
Tessellations
A tessellation covers a surface with shapes and no gaps or overlaps, so the angles that meet at each vertex must add to exactly 360°. A regular shape tessellates on its own only if its angle divides 360° exactly: triangles (60°, six fit), squares (90°, four fit) and hexagons (120°, three fit). Pentagons (108°) leave a gap. To find a missing shape, subtract the angles already there from 360°. A pattern is made by moving a shape in one of three ways: a translation slides it, a reflection flips it like a mirror, and a rotation turns it about a point.
Sample questions
Regular triangles are fitted around a vertex with no gaps. Each angle of a regular triangle is 60°. 4 are already in place. How many more are needed to fill the vertex?
- The angles at a vertex fill a full turn of 360°.
- 4 × 60° = 240°, so 120° is still empty.
- 120 ÷ 60 = 2, so 2 more triangles are needed.
Answer: 2
Each angle of a regular square is 90°. Angles at a vertex add to 360°. How many regular squares meet at a vertex?
- In a tessellation there are no gaps and no overlaps, so the angles at a vertex fill a full turn of 360°.
- Each square contributes 90°.
- 360 ÷ 90 = 4, so 4 squares meet at a vertex.
Answer: 4
Each angle of a regular hexagon is 120°. Angles at a vertex add to 360°. How many regular hexagons meet at a vertex?
- In a tessellation there are no gaps and no overlaps, so the angles at a vertex fill a full turn of 360°.
- Each hexagon contributes 120°.
- 360 ÷ 120 = 3, so 3 hexagons meet at a vertex.
Answer: 3