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How to Rotate a Shape (Mathematics Lesson)

How to Rotate a Shape

To rotate a shape, rotate each point on the shape by the angle of rotation about the center of rotation.

Each point on the shape above has been rotated θ degrees clockwise about the center of rotation (x, y).

Common Examples of How to Rotate a Shape

Common rotations are rotations of 90°, 180°, 270° and 360° about the origin.

It is often possible to rotate these shapes by eye.

Read more about common rotations

A Real Example of How to Rotate a Shape

It is possible to rotate a shape by any angle about any center of rotation.

Question: Rotate the shape below by 60° clockwise about the point (3, 1).

Step 1: Find the center of rotation.
The center of rotation has Cartesian coordinates (3, 1).
It is 3 units along the x-axis and 1 unit up the y-axis.

For each vertex (corner) of the shape (vertex A in this example).

Step 2: Draw a line from the center of rotation to the vertex.

Step 3: Measure the angle of rotation from the line.
Using a protractor, find 60° clockwise from the line found.

Step 4: Draw a line of the same length as in Step 2 from the center of rotation at the angle found in Step 3.

Vertex A has been rotated 60° about (3, 1) to find A', the corresponding point on the rotated shape.

Repeat for vertex B and C:

With all vertices of the shape rotated, the rotate shape can be drawn:

The shape has been rotated 60° clockwise about (3, 1).

Geometry Lessons
Interactive Test

A rotation turns a shape around a center.

A rotation is a type of transformation.


A rotation is described by the angle the shape turns about a center of rotation.

  • The center of rotation is the point that a shape rotates about.

    It can be described by Cartesian coordinates, (x, y).

    The center of rotation may be found by observation or by construction.

  • The angle of rotation is the angle that the shape has been rotated about.

    It can be described in degrees or radians. The direction of the rotation (clockwise or counterclockwise) can also be described.
Read more about how to describe a rotation


The center of rotation can be described using Cartesian coordinates, (x, y).

  • The co-ordinate on the left is the x-coordinate.

    It describes how far along the x-axis, or how far across, the point is.

  • The co-ordinate on the right is the y-coordinate.

    It describes how far up the y-axis, or how far up, the point is.
For example, the point below is 2 along the x-axis and 3 up the y-axis. Therefore its Cartesian coordinates are (2,3)


The direction of rotation is needed to describe a rotation.

  • If the rotation is in the same direction as the hands of a clock, the direction is clockwise.

  • If the rotation is in the opposite direction as the hands of a clock, the direction is counter-clockwise or anti-clockwise.

Top Tip


Imagine a shape is drawn on a sheet of paper...

Imagine sticking a pin through the paper and into a surface.

If you span the paper around, the pin would stay in place and every other point on the paper would turn in a circle around it.

The pin would be the center of rotation, and the amount you span the paper would be the angle of rotation.