A rotation is a transformation where every point moves through the same angle about a fixed point − called the centre of rotation.

Location of Centre


R is a rotation of Y10_Rotation_01.gif about centre of rotation O.Y10_Rotation_02.gif

  • R: A Y10_Rotation_03.gif A´ or R(A) = A´
  • The angle of rotation is the angle between the line joining the centre to a point, and the line joining the centre to the image of the point.
  • For anti-clockwise rotations, the angle of rotation is positive.
  • For clockwise rotations the angle is negative.


Properties of Rotation

Triangle PQR maps to triangle P´Q´R´ under a rotation of Y10_Rotation_04.gif°, centre O.



  • All points and lines, except the centre, turn through an angle of °

    e.g. Y10_Rotation_06.gifPOP´ = Y10_Rotation_06.gifQOQ´ = Y10_Rotation_06.gif ROR´ = q

  • Length, angle size and area are invariant.

    e.g. length PQ = length P´Q´

    • Y10_Rotation_06.gifPQR = Y10_Rotation_06.gifP´Q´R´

      area Y10_Rotation_07.gif PQR = area Y10_Rotation_07.gif P´Q´R´

  • The centre of rotation is the only invariant point.
  • Rotation is a direct transformation.

    i.e. PQR and P´Q´R´ are both labelled in the anti-clockwise direction.

  • Rotation is an isometry.

    i.e. Y10_Rotation_07.gif PQR is congruent to Y10_Rotation_07.gifP´Q´R´.

See examples of rotations − button_animation.gif

rotation.jpgLocation of Centre of Rotation

Given a figure and its image, to find the centre of the rotation:

1. Join any point and its image, and draw the perpendicular bisector of this line.

2. Repeat for another point and its image.

3. The centre of rotation is the intersection of the mediators (the perpendicular bisectors).

Download an interactive spreadsheet (Microsoft Excel) showing reflections, rotations, translations and enlargements.


(Windows users, right click and "Save target as..." to save the files on your computer.)