Rotation

In a Nutshell

A rotation turns a shape around a fixed point called the centre of rotation, by a given angle and direction (clockwise or anticlockwise).

To describe a rotation fully you need three pieces of information:

  1. The centre of rotation (the fixed point).
  2. The angle of rotation (e.g. 90°90°, 180°180°, 270°270°).
  3. The direction — clockwise or anticlockwise.

For rotations about the origin:

  • 90°90° clockwise: (x,y)→(y,−x)(x, y) \to (y, -x)
  • 180°180°: (x,y)→(−x,−y)(x, y) \to (-x, -y)
  • 90°90° anticlockwise (or 270°270° clockwise): (x,y)→(−y,x)(x, y) \to (-y, x)

The shape and its image are always congruent — the same size and shape, just turned.

Explore: rotation

Rotation animator with arc trails A shape on a coordinate grid rotates around a marked centre point. Each vertex traces a visible arc to show that rotation is circular movement.

Choose an angle and press Transform to see the rotated image. The centre of rotation is the origin.

Watch it work

Question: Rotate the point P(2,3)P(2, 3) by 90°90° clockwise about the origin.

Have a go

Q1. Rotate (1,4)(1, 4) by 180°180° about the origin.

Q2. Rotate (−3,2)(-3, 2) by 90°90° clockwise about the origin.

Q3. Rotate (5,−1)(5, -1) by 90°90° anticlockwise about the origin.

Q4. A triangle has vertices A(1,1)A(1, 1), B(3,1)B(3, 1) and C(3,4)C(3, 4). Rotate it 180°180° about the origin and write the image coordinates.