Lines of symmetry in shapes

Each regular shape has its own number of lines of symmetry. Some have one, some have many. Let's go through the most common shapes.

What is this?

Square , 4 lines of symmetry. Two through the middle (vertical and horizontal), and two through the diagonals. Each fold gives two halves that match.

Rectangle , 2 lines of symmetry. The vertical and horizontal lines through the middle. The diagonals do not work , folding along a diagonal does not make the halves match (try it!).

Equilateral triangle , 3 lines of symmetry. Each goes from a corner to the middle of the opposite side.

Isosceles triangle (with two equal sides) , 1 line of symmetry. The line from the top corner to the middle of the bottom side.

Scalene triangle (all sides different) , 0 lines of symmetry.

Rhombus , 2 lines of symmetry. Along the two diagonals.

Circle , infinite. Every line through the centre divides the circle into matching halves.

Regular pentagon (5 equal sides) , 5 lines of symmetry.

Regular hexagon (6 equal sides) , 6 lines of symmetry.

The pattern is clear: a regular polygon with $n$ sides has $n$ lines of symmetry.

Why does a rectangle (which has 4 right angles, like a square) have only 2 lines of symmetry, while a square has 4? Because the sides of a rectangle are not all equal. When you fold along the diagonal, the two halves are mirror copies in shape but not in their corner positions , they don't match.

Worked examples

Example 1. How many lines of symmetry in a square?

• 4: two through midpoints of sides, two through diagonals.

Example 2. How many lines of symmetry in a regular octagon?

• A regular octagon has 8 equal sides.
• 8 lines of symmetry.

Example 3. Does a parallelogram (not a rectangle) have any lines of symmetry?

• Try vertical fold , no match.
• Try horizontal fold , no match.
• Try diagonals , no match.
• 0 lines of symmetry.

Try it

1. How many lines of symmetry does a regular pentagon have?
2. Does an isosceles trapezium have line symmetry? How many lines?
3. Does the letter "M" have any line symmetry? How many?
4. How many lines of symmetry does a kite have?
5. Draw a shape that has exactly 2 lines of symmetry.

Hands-on

Paper-fold shapes. Cut out a paper square. Fold it carefully to discover all 4 lines of symmetry. Now do the same with a rectangle and confirm there are only 2 lines.

Letter check. Write all the capital letters A to Z. For each, decide how many lines of symmetry. Make a table.