Rotations –

Rotations are one of the four transformations tested in GCSE Maths across AQA, Edexcel, and OCR. A rotation turns a shape around a fixed point called the centre of rotation by a specified angle in a given direction. This guide explains how to perform and describe rotations, covering 90°, 180°, and 270° turns with worked examples from Foundation to exam style.

What Is a Rotation?

A rotation turns every point of a shape through the same angle about a fixed point. The shape does not change size or shape — it only changes position and orientation.

Key Formulas

To describe a rotation fully you must state three things: the centre of rotation, the angle of rotation, and the direction (clockwise or anticlockwise). For 180°, direction is not required since clockwise and anticlockwise give the same result.

Step-by-Step Method

1. Identify the centre of rotation from the question or diagram.
2. Use tracing paper. Trace the shape, place your pencil on the centre of rotation, and turn the tracing paper by the given angle.
3. Plot the rotated vertices and draw the image.
4. To describe a rotation, find the centre by trial using tracing paper, then state the angle and direction.

Worked Example 1 — Foundation Level

Question: Rotate the triangle with vertices A(1, 2), B(3, 2), and C(3, 4) by 90° clockwise about the origin.

Working:

90° clockwise about (0, 0): (x, y) maps to (y, −x).

A(1, 2) maps to A'(2, −1)

B(3, 2) maps to B'(2, −3)

C(3, 4) maps to C'(4, −3)

Answer: The image has vertices A'(2, −1), B'(2, −3), C'(4, −3).

Worked Example 2 — Higher Level

Question: Rotate the point P(−2, 5) by 180° about the origin.

Working:

180° about (0, 0): (x, y) maps to (−x, −y).

P(−2, 5) maps to P'(2, −5).

Answer: P' = (2, −5).

Worked Example 3 — Exam Style

Question: Triangle A is mapped to triangle B by a rotation. Vertex (1, 3) on A maps to (3, −1) on B. The centre of rotation is the origin. Describe the rotation fully.

Working:

(1, 3) maps to (3, −1). Using the rule (x, y) maps to (y, −x), this matches a 90° clockwise rotation about the origin.

Check: x = 1, y = 3. (y, −x) = (3, −1). Correct.

Answer: Rotation, 90° clockwise, centre (0, 0).

Common Mistakes

• Not stating all three parts of the description. You must give the centre, the angle, and the direction. Missing any one of these loses marks.
• Confusing clockwise and anticlockwise. Clockwise follows the direction of a clock's hands. If you confuse them, the image ends up in the wrong quadrant.
• Using the wrong centre. If the centre of rotation is not the origin, the coordinate rules above do not apply directly. Use tracing paper instead.

Exam Tips

• Use tracing paper in your exam — it is allowed and makes rotations much easier to perform accurately.
• For 180° rotations, you do not need to state a direction since both clockwise and anticlockwise produce the same result.
• A 270° clockwise rotation is the same as a 90° anticlockwise rotation. Use whichever is simpler.
• When finding the centre of rotation, try the origin first. If that does not work, look for a point equidistant from a vertex and its image.
• Always check your image by confirming that each image vertex is the same distance from the centre as the corresponding original vertex.

Practice Questions

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

Q2 (Foundation): Rotate the point (2, 5) by 90° anticlockwise about the origin.

Q3 (Higher): A shape is rotated so that the point (3, 1) maps to (−1, 3). The centre is the origin. Describe the rotation fully.

Practise rotations with instant feedback free on GCSEMathsAI.

Related Topics

• Reflections
• Translations and Column Vectors
• Enlargement and Centre of Enlargement
• Transformations — Reflection, Rotation, Translation

Summary

• A rotation turns a shape about a fixed centre by a given angle in a specified direction. To describe a rotation you must state the centre, the angle, and the direction (except for 180°). Know the coordinate rules for rotations about the origin: 90° clockwise maps (x, y) to (y, −x), 90° anticlockwise maps (x, y) to (−y, x), and 180° maps (x, y) to (−x, −y). Use tracing paper in exams to perform and check rotations.