Symmetry (including Reflection and Rotation)
182. The Circle's Infinite Mirror · every diameter splits it perfectly in half
Reflecting any point on the circle across a diameter lands exactly on the circle again.
A circle has infinitely many lines of symmetry — every diameter is a line of symmetry. Reflecting any point on the circle across any diameter lands you back on the circle. Maximum possible symmetry for a 2D figure.
What this lesson covers
Try to break it
Drag P around the circle to rotate the axis of symmetry; drag Q anywhere on the circle. Its mirror Q' always lands on the circle on the other side of OP, equidistant from the axis. When Q sits exactly on OP, Q' coincides with Q — the point is its own reflection. Any diameter works as a mirror.
How you build it
Draw a circle with a diameter and a reflected point.
- Draw a circle with centre O.
- Draw any line passing through O.
- Place a point Q on the circle.
The proof, step by step
Prove that reflecting any point of the circle across a diameter lands on the circle.
- Let the line through O be L. Since L passes through the centre, it is a diameter.
- Reflection across L maps any point X to X' such that L is the perpendicular bisector of XX'.
- Since O lies on L, the distance OX equals OX'. Thus X' also lies on the circle.
- Therefore, L divides the circle into two congruent halves, making it a line of symmetry.
Worked example
A circle has an infinite number of lines of symmetry. How many lines of symmetry does a square possess?
A square has 4 lines of symmetry: 2 diagonals and 2 lines joining midpoints of opposite sides.
- 2
- 4 — correct
- 6
- Infinite