Topics · Primary Maths · Symmetry & Tessellation
Symmetry and tessellation
A shape is symmetric about a line if folding along it makes the two halves land exactly on each other. A shape tessellates if copies of it tile a surface with no gaps and no overlaps.
- 200 exam papers analysed
- appears in 63% of P6 Maths papers
- ~3 min read
01 The fold test
A line of symmetry is a fold line. Fold the shape along it and the two halves must match exactly — every corner onto a corner, every edge onto an edge.
That is the test, and it settles every case. A line that merely cuts a shape into two pieces of the same area is not a line of symmetry unless the pieces also have the same shape and land on each other.
02 How many lines a shape has
Counting them is a standard question, and the answers are worth knowing rather than working out each time.
- Square — 4. Two through the middles of opposite sides, two along the diagonals.
- Rectangle — 2. Through the middles of the sides only. Its diagonals are not lines of symmetry, which is the answer most often got wrong.
- Equilateral triangle — 3. One from each corner to the middle of the opposite side.
- Isosceles triangle — 1. Down the middle, between the two equal sides.
- Circle — endless. Every line through the centre works.
- Parallelogram — none, unless it happens to be a rhombus or a rectangle.
03 Tessellation
A shape tessellates if copies of it cover a flat surface with no gaps and no overlaps, carrying on for ever. Turning and flipping the copies is allowed.
Squares, rectangles, triangles and hexagons all tessellate. Circles do not — however you pack them there are always gaps between.
The reason is angles: at every point where corners meet, they must add to exactly 360°. Six equilateral triangles do it, because 6 × 60° = 360°. A regular pentagon cannot, because 108° will not divide into 360°.
04 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- symmetrysaySIM-uh-tree
The property of matching exactly when folded along a line, or turned about a point.
- line of symmetrymore than onelines of symmetry
A fold line where the two halves of a shape land exactly on each other.
- symmetric figuresaysi-MET-rik
A shape that has at least one line of symmetry.
- tessellatesayTESS-uh-layt
To fit copies of a shape together covering a surface with no gaps and no overlaps.
- tessellationsaytess-uh-LAY-shuhnmore than onetessellations
The pattern made when a shape tessellates.
05 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P4
Finding lines of symmetry, and completing a symmetric figure on a grid.
Practise P4 Symmetry & Tessellation — 10 questions →61% of P4 papers
- P5
Symmetry of the standard shapes, and drawing the other half of a figure.
Practise P5 Symmetry & Tessellation — 10 questions →17% of P5 papers
- P6
Symmetry inside composite figures, and tessellating patterns.
Practise P6 Symmetry & Tessellation — 10 questions →63% of P6 papers
06 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“A rectangle has 4 lines of symmetry, like a square.”
Two. Its diagonals are not fold lines — fold a long thin rectangle along its diagonal and the halves stick out past each other.
“Any line that cuts a shape in half is a line of symmetry.”
Only if the halves land on each other when folded. Equal areas are not enough.
“A parallelogram has 2 lines of symmetry.”
An ordinary parallelogram has none. It has rotational symmetry, which is a different thing.
“Circles tessellate because you can pack them tightly.”
Tightly is not the test. However you arrange them there are gaps left between, so they do not tessellate.
“Every shape tessellates if you turn it enough.”
The corners meeting at a point must total 360°. A regular pentagon's 108° will not, so no arrangement works.
Now try it
30 questions across 3 levels, marked as you go with the working shown.