Lines of symmetry
Not every shape has an axis of symmetry, and the ones that do can have one, two, three or even more. Learning to spot them and count them is the heart of Year-4 symmetry.
Two tests for any candidate line
Before you draw a line and say "this is an axis of symmetry", check both:
- Half-and-half — does the line split the shape into two halves of the same size and the same outline?
- Mirror match — if you mentally fold along the line, do the two halves land exactly on top of each other?
A diagonal of a rectangle passes the first test (it splits the rectangle into two equal triangles) but fails the second test (the triangles don't match when you fold along it). So a diagonal is not an axis of symmetry of a rectangle — unless the rectangle is a square.
Counting axes — by shape
Triangles
- Scalene (all sides different): 0 axes.
- Isosceles (two sides equal): 1 axis — from the top vertex to the middle of the base.
- Equilateral (all sides equal): 3 axes — one from each vertex.
Quadrilaterals
| Shape | Axes | Where |
|---|---|---|
| general quadrilateral | 0 | — |
| rectangle (not square) | 2 | through the midpoints of opposite sides |
| rhombus (not square) | 2 | along the two diagonals |
| square | 4 | both midline pairs + both diagonals |
| isosceles trapezium | 1 | between the two parallel sides |
| parallelogram (not rhombus) | 0 | — |
A parallelogram that is not a rhombus and not a rectangle has no axes of symmetry — even though it looks "balanced", folding it never works.
Regular polygons
A regular polygon has all sides equal and all angles equal. It always has as many axes as sides:
- regular pentagon (5 sides): 5 axes
- regular hexagon (6 sides): 6 axes
- regular heptagon (7 sides): 7 axes
For polygons with an even number of sides, the axes pair up: half of them go through opposite vertices, half through midpoints of opposite sides. For an odd number of sides, every axis goes through one vertex and the midpoint of the opposite side.
Circle
A circle has infinitely many axes of symmetry — every line through the centre works. That's why circles look "perfect" in any direction.
Drawing the axes step by step
- Look at the shape and ask: where are the matching pairs of corners or sides?
- Draw a line that passes between each matching pair, hitting the midpoint of any side it crosses.
- Use the two tests above to confirm.
For tricky shapes, trace the figure on tracing paper, fold it, and watch what happens. The fold that makes it match is the axis.
A worked example
How many axes of symmetry does the capital letter H have?
The letter H has:
- a horizontal line through the middle (the top half is the mirror of the bottom half), and
- a vertical line through the middle (the left vertical bar is the mirror of the right).
That's 2 axes of symmetry.
What about the letter O (drawn as a perfect circle)?
If you draw O as a circle, it has infinitely many axes. If you draw it as a tall oval, it has 2 — one horizontal and one vertical.
What's next
- Completing a symmetric figure — drawing the missing half
- Symmetry around us
- Back to the introduction