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.
Explore this idea with maths tools
Maths you can move
Build 8 + 5
Tap two gold counters to fill the ten-frame.
8 in the frame + 5 waiting. The total stays the same.
Words to know
- tessellation
- a pattern of shapes that covers a surface with no gaps and no overlaps
- vertex
- a corner where shapes meet
- translation
- a slide
- reflection
- a flip, like an image in a mirror
- rotation
- a turn about a point
Worked examples
Each angle of a regular hexagon is 120°. How many hexagons meet at a vertex of a tessellation?
- 1.The angles at a vertex add to 360°.
- 2.360 ÷ 120 = 3.
- 3.3 hexagons meet at a vertex.
= 3
Two regular hexagons (120° each) and some regular triangles (60° each) meet at a vertex. How many triangles meet there?
- 1.The hexagons use 2 × 120 = 240°.
- 2.That leaves 360 − 240 = 120° for the triangles.
- 3.120 ÷ 60 = 2.
= 2
Three regular pentagons (108° each) are fitted around a vertex. How many degrees are left as a gap?
- 1.Three pentagons use 3 × 108 = 324°.
- 2.A full turn is 360°, so the gap is 360 − 324 = 36°.
- 3.A fourth pentagon needs 108°, so it does not fit: pentagons do not tessellate on their own.
= 36
Shape 2 is a mirror image of shape 1, so it faces the opposite way. Which transformation moved shape 1 onto shape 2?
- 1.A slide keeps the shape facing the same way, so it is not a translation.
- 2.A turn rotates the shape without reversing it, so it is not a rotation.
- 3.A mirror image is a reflection.
= Reflection
For parents
Method: Angles at a vertex add to 360
What to say: How many degrees are there in a full turn? How many degrees do the shapes already use? What is left?
Common mistakes:
- Divides 180 instead of 360 because a straight line is the angle most students remember.
- Gives the number of sides of the shape instead of the number of shapes that fit around a vertex.
- Gives the angle of the shape as the answer, instead of dividing 360° by it.
- Leaves out the shape that is already drawn at the vertex, so counts one too few.
- Thinks regular pentagons tessellate because they look like they fit together in a pattern.
- Thinks regular octagons tessellate because they appear on tiled floors.
- Thinks a shape with more sides tessellates because it looks rounder or fuller.
- Calls every pattern change a slide (translation), ignoring flips and turns.
- Calls a turn (rotation) or a slide a reflection, because the shape ends up pointing a different way.
- Calls a flip or a slide a rotation, because the shapes sit in different places or positions.
- Divides the whole 360° by the new shape's angle, ignoring the shapes already at the vertex.
- Takes away the angle of one shape from 360° when two or more shapes are already there.
- Gives the total number of shapes at the vertex instead of the number of the second shape.
- Gives the total angle used by the shapes instead of the gap that is left over.
- Gives the exterior angle (180° minus the interior angle) as the size of the gap.
- Takes only one shape's angle from 360° when several shapes are at the vertex.
- Gives the angle left over, in degrees, instead of the number of shapes that fit in it.
- Gives the first shape again as the missing third shape.
- Gives the second shape again as the missing third shape.
- Guesses a pentagon without checking the angle, because it is the shape between the square and the hexagon.