# How to Describe a Rotation

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

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

# The 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

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.

The angle of rotation can be found by drawing lines from the center of rotation to corresponding points on the object and image, and measuring the angle between the lines.

# Real Examples of How to Describe a Rotation

Rotations of 90°, 180°, and 270°, often about the origin, are the most commonly asked for in examinations.

It is possible to describe a rotation of any angle about any point, although it is harder to see the angle and center just by looking.

Question: Describe the rotation below.

The light blue shape has been rotated 60° about (1, 1).

Geometry Lessons
show

# HOW TO THINK OF THE CENTER OF ROTATION

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.

##### Note
WHAT IS A ROTATION?

A rotation turns a shape around a center.

A rotation is a type of transformation.

# HOW TO DESCRIBE THE CENTER OF ROTATION

The center of rotation can be described using Cartesian co-ordinates, (x, y).

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

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

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

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 co-ordinates are (2,3)

CLOCKWISE AND COUNTER-CLOCKWISE

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.