Rotation - KS3 Maths - BBC Bitesize

Key points

An image of a partial rotation of a shape by one hundred and eighty degrees. — Image caption,
For a rotation of 180° it does not matter if the turn is clockwise or anti-clockwise, as the outcome is the same.

A is one of the four types of .

A rotation has a turning effect on a shape. The result is a shape.
Rotation turns a shape around a fixed point called the . This point can be inside the shape, a of the shape or outside the shape.
Rotations can be or and a multiple of 90° (90°, 180° or 270°) is used.
To understand rotations, a good understanding of angles and rotational symmetry can be helpful.

To fully describe a rotation, it is necessary to specify the angle of rotation, the direction, and the point it has been rotated about.

Rotating about the centre of a shape

When a shape about a central point the and the will always overlap.

A piece of tracing paper can be useful when plotting a rotation.

To work out the position of the new shape after a rotation:

Place the tracing paper over the shape.
Copy the shape on to the tracing paper.
Rotate the tracing paper in the correct direction and the specified angle, keeping the centre of rotation fixed.
The tracing paper now shows the position of the new shape.
Remove the tracing paper and draw the shape in its new location.

Examples

Skip image gallery

Image caption,
Rectangle ABCD is to be rotated 90° clockwise about the centre of rotation, P.
Image caption,
Place the tracing paper over the shape and centre of rotation.
Image caption,
Trace the shape with a pencil and mark the centre of rotation.
Image caption,
Turn the tracing paper 90° clockwise keeping the centre of rotation fixed on the same point, P. This shows where the new shape’s location is.
Image caption,
Draw the shape in its new location. The new shape can be labelled with similar letters to explain where each vertex has been rotated to. Rectangle A'B'C'D' is a 90° clockwise rotation about point, P.
Image caption,
Rotations can also occur on sets of axes. This problem can be approached in the same way. Trapezium ABCD is to be rotated 180° about the centre of rotation, P, with co-ordinate (5, 4).
Image caption,
Place the tracing paper over the shape and centre of rotation. Trace the shape with a pencil and mark the centre of rotation.
Image caption,
Turn the tracing paper 180° keeping the centre of rotation on the same fixed point, P. For a rotation of 180° it does not matter if the turn is clockwise or anti-clockwise as the outcome is the same. This shows where the new shape’s location is.
Image caption,
Draw the shape in its new location. The new shape can be labelled with similar letters to explain where each vertex has been rotated to. Trapezium A'B'C'D' is a 180° rotation about point, P.
Image caption,
To fully describe a rotation, it is necessary to specify the angle of rotation, the direction, and the point it has been rotated about. Triangle A'B'C' is a rotation 90° anti-clockwise (or 270° clockwise) about point P.

Question

The orange L-shape has been rotated about point P. What angle and direction has it been rotated by?

An image of a square grid. The grid has a length of fourteen squares and a width of ten squares. An L shape has been drawn on the grid. Starting from the top left vertex, the shape has sides of length two squares to the right, four squares down, two squares to the right, two squares down, four squares to the left and six squares up. A point, one square to the right and three squares below the top left vertex, has been labelled, point P. A second L shape has been drawn to represent a rotation. This L shape has a vertex two squares to the left and two squares below the top left vertex of the original shape. The shape has sides six squares to the right, two squares down, four squares to the left, two squares down, two squares to the left and four squares up. The original shape is coloured orange. The second shape is coloured blue.

Rotating about other centre of rotations

The location of the centre of affects the final position of the image. If the centre of rotation is a of the object, then that point will remain unchanged through the rotation.

It is possible to plot a rotation without tracing paper.

To work out the position of the new shape after a rotation:

Imagine a line from the centre of rotation to a vertex on the shape.
Rotate this line in the correct direction and the specified angle. keeping the centre of rotation fixed.
The end of this line will to the new location of that vertex.
Repeat the process for further vertices on the shape.

Examples

Skip image gallery

Image caption,
Rotate the flag shape 180° about the centre of rotation, P.
Image caption,
Choose a vertex on the shape, for example A. Imagine a line from the centre of rotation, P, to the vertex marked A.
Image caption,
Rotate the line segment AP 180°, keeping the centre of rotation P fixed. For a rotation of 180° it does not matter if the turn is clockwise or anti-clockwise as the outcome is the same. The end of this line, A', is the new position of point A.
Image caption,
Choose another vertex B and imagine a line from P to B. Rotate this line 180°. The end of this line is the new position of point B.
Image caption,
Once this has been done for enough vertices it is possible to draw the whole image in its correct position. This can be checked by using tracing paper, if required.
Image caption,
Triangle ABC is to be rotated 90° anti-clockwise about vertex C. Since vertex C is the centre of rotation it will remain unchanged.
Image caption,
Imagine a line segment AC rotated 90° anti-clockwise, keeping the centre of rotation C fixed. The end of this line is the new position of point A. A has rotated to (1, 5).
Image caption,
Imagine line segment BC rotated 90° anti-clockwise, keeping the centre of rotation C fixed. The end of this line is the new position of point B. B has rotated to (5 , 5).
Image caption,
With all three vertices now plotted the rotated shape can be completed. The image is a rotation, 90° anti-clockwise about vertex C.
Image caption,
The location of the centre of rotation affects the final position of the shape. Each of these triangles is a rotation of triangle A, 90° clockwise. B is a rotation 90° clockwise about point P. C is a rotation 90° clockwise about point Q. D is a rotation 90° clockwise about point R. Notice the orientation of each solution is the same but the position is different.

Question

The ABCD is rotated 90 clockwise about point D. Which coordinate is point A rotated to?

The image shows a set of axes. The horizontal axis is labelled x. The values go up in ones from zero to eight. The vertical axis is labelled y. The values go up in ones from zero to eight. Parallelogram A B C D has been plotted and has vertices with coordinates, A equals two comma seven, B equals six comma seven, C equals five comma five, and D equals one comma five.

Practise rotation and rotating a shape about a given point

Quiz

Practise rotation and how to rotate a shape about a given point in this quiz. You may need a pen, paper and tracing paper for this quiz.

Real-life maths

An image of a gymnast on a pommel horse. — Image caption,
A gymnast on a pommel horse, performing a rotation move called circles.

A gymnast is required to perform complicated tumbling when competing on a variety of apparatus or as a floor exercise. Many of these routines involve acrobatics where the competitor is rotating.

For example, on the pommel horse a gymnast can perform a move called circles. Circles requires the competitor to rotate or swing their legs around the apparatus whilst alternating their hand positions on the pommels (handles). Some events require an elaborate dismount. This could include a somersault where a person’s body rotates 360°.