Fold it flat

The idea

Every diameter of a circle is a line of reflection symmetry of the circle.

Folding a circle

Take a circular paper and fold it so that the boundaries overlap. The crease you see is a line of reflection symmetry of the circle.

Does the crease pass through the centre of the paper?

• Yes, always
• No, it can be anywhere
• Only sometimes

Fold the paper

Drag P and Q to choose a crease. The piece without the centre is folded over. The shape on top shows where it lands.

Folds that work

Every diameter of a circle is a line of reflection symmetry of the circle.

Fold along a diameter and the two halves land exactly on each other. Fold along any other chord and the folded edge falls inside the circle. So a crease that makes the boundaries overlap always passes through the centre.

Notes

Every diameter of a circle is a line of reflection symmetry of the circle.

Check yourself

A circular paper is folded so that the boundaries overlap exactly. Where does the crease pass?

How many lines of reflection symmetry does a circle have?

A circle is folded along a chord AB that does not pass through the centre. What happens?

A circle has radius 5 cm. A crease that makes the boundaries overlap runs across the whole circle. How long is that crease, in cm?

Answer: 10 cm

The crease is a diameter: from the circle, through the centre, to the circle again. That is 5 + 5 = 10 cm.

• Through the centre of the circle — correct. Yes. The crease is a diameter.
• Through any two points of the circle. A chord that misses the centre does not work: the folded edge falls inside the circle.
• Through the midpoint of a radius. A crease that misses the centre does not make the boundaries overlap.
• Only 2: one across and one up and down. Those are two of them, but every other diameter works too.
• Infinitely many: one for every diameter — correct. Yes. Every diameter is a line of symmetry, and a circle has infinitely many diameters.
• Only 4, like a square. A square has 4 lines of symmetry. A circle has a line for every diameter.
• The two parts still match exactly. They match only when the crease passes through the centre.
• The paper cannot be folded. It can be folded, but the edges will not meet.
• The folded edge falls inside the circle, so the boundaries do not overlap — correct. Yes. Only a crease through the centre makes the boundaries overlap.