Multiple lines of symmetry

Some shapes have just one line of symmetry. Others have several. And a circle has infinitely many. Counting the lines of symmetry of a shape is a quick way to capture how "balanced" it is.

Concept

Here is a table of how many lines of symmetry common shapes have:

A few interesting observations:

• For regular polygons with $n$ sides, the number of lines of symmetry is always $n$. Each line passes either through a vertex and the midpoint of the opposite side (for odd $n$) or through two opposite vertices and through two opposite edge midpoints (for even $n$).
• The circle is the most symmetric shape in the plane. Every line through its centre is a line of symmetry.
• Adding constraints (like equal sides or equal angles) usually increases symmetry. A general quadrilateral has $0$ lines; a rectangle has $2$; a square has $4$.

Finding all the lines of symmetry of a polygon

For a regular polygon with $n$ sides and centre $O$:

• If $n$ is odd: each line of symmetry passes through one vertex and through the midpoint of the opposite side. That gives $n$ lines (one per vertex).
• If $n$ is even: half of the lines pass through two opposite vertices. The other half pass through the midpoints of two opposite sides. Total still $n$.

For irregular shapes, the only way is to check each candidate fold direction. Cutting the shape out and folding is the most reliable test.

A subtle point

A diagonal of a rhombus is a line of symmetry. A diagonal of a rectangle is not a line of symmetry (it doesn't fold cleanly because the sides have different lengths). Only when the diagonals are perpendicular and equal in length , i.e., for a square , do both diagonals work as lines of symmetry.

Worked examples

Example 1. How many lines of symmetry does a square have? Sketch them.

• $4$ lines:
  • Two through the midpoints of opposite sides (horizontal and vertical).
  • Two through opposite corners (the two diagonals).

Example 2. How many lines of symmetry does a regular octagon ($8$-gon) have? Sketch a few.

• $8$ lines:
  • $4$ through pairs of opposite vertices.
  • $4$ through midpoints of opposite sides.

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

• No. None of its sides or diagonals work.

Example 4. How many lines of symmetry does a five-pointed star have (the regular pentagram drawn as a five-pointed star)?

• $5$ lines, each going through one point of the star and through the centre to the midpoint of the opposite edge.

Try it yourself

1. Sketch all $4$ lines of symmetry of a square.
2. How many lines of symmetry does an isosceles trapezium have?
3. How many lines of symmetry does a rhombus (not a square) have? Where are they?
4. How many lines of symmetry does a regular dodecagon ($12$-gon) have?
5. Look at the Indian flag. Does it have a line of symmetry? Where?
6. Sketch a six-petalled flower. How many lines of symmetry?
7. How many lines of symmetry does the letter "X" have?
8. Investigate: a quadrilateral with exactly one line of symmetry , name it. (Hint: kite or isosceles trapezium.)

Activity

Paper-fold polygons. Fold a square piece of paper in half. Fold again. Fold again at $45°$ (corner to corner). Now use scissors to cut a design into the folded edge. Open the paper. The design will have all the symmetries of a square , $4$ lines and rotational symmetry too. Try starting with a paper folded into thirds for a different count of symmetries.