What Does it Mean to Rotate a Shape?

Rotating a shape means to change its direction by turning it. The size and shape do not change during rotation. A shape is often rotated about a specific point called the centre of rotation. We also need to know what angle and direction the shape is rotated in.

For example, this triangle has been rotated 90° counter-clockwise about the point.

You can tell that a shape has been rotated because it is not facing the same direction as it was originally. The original shape is called the object and the rotated shape is called the image.

When rotated, a shape remains the same distance away from the centre of rotation. It is just in a different direction.

Rules for Rotating a Shape About the Origin

The rules for rotating shapes using coordinates are:

How to Rotate a Shape by 90 Degrees

To rotate shape 90° clockwise about the origin, all original coordinates (x, y) becomes (y, -x). To rotate a shape 90° counter-clockwise about the origin, the coordinates (x, y) become (-y, x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.

For example, use the rule (x, y) to (y, -x) to rotate the shape 90° clockwise.

To use this rule, simply switch the (3, 1) to (1, 3) and then make the 3 negative to get (1, -3).

How to Rotate a Shape by 180 Degrees

To rotate a shape by 180° clockwise or counter-clockwise, the rule is to replace the (x, y) coordinates with (-x, -y). For example, a coordinate at (3, 1) will move to (-3, -1) after a 180° rotation.

Simply multiply each coordinate by -1 to rotate a shape 180°.

If a coordinate is negative, it will become positive after a 180° rotation. For example, the coordinate (-1, -4), will move to (1, 4) after a 180° rotation.

How to Rotate a Shape by 270 Degrees

To rotate shape 270° clockwise about the origin, all original coordinates (x, y) becomes (-y, x). To rotate a shape 270° counter-clockwise about the origin, the coordinates (x, y) become (y, -x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.

For example, to rotate the point (3, 1) 270° clockwise, it becomes (-1, 3). Simply switch the x and y coordinates to get (1, 3) and then multiply the 1 by -1 to get (-1, 3).

How to Rotate a Shape Using Tracing Paper

To rotate a shape using tracing paper:

1. Place the tracing paper over the shape and draw around the shape.
2. Draw an arrow from the centre of rotation pointing upwards.
3. Keep the pen over the centre of rotation and rotate the tracing paper.
4. Stop when the arrow is facing either right (for 90° CW / 270° CCW turn), down (for 180° turn) or left (for 270° CW / 90° CCW turn).
5. Draw the shape in this new position below the tracing paper.

The easiest way to rotate a shape is to use tracing paper.

For example, use tracing paper to rotate the shape 90° clockwise about the point.

After a 90° clockwise rotation, the upwards arrow is facing right. Use the grid lines of the paper to help line up the arrow correctly, ensuring that it is completely horizontal.

Then draw in the shape below.

Here is another example. Rotate the shape 270° clockwise about the point using tracing paper.

Every lot of 90° is equivalent to a quarter turn. 270° is 3 lots of 90° and so, 270° is equivalent to three-quarters of a turn.

The upward-facing arrow will be facing to the left after a 270° clockwise rotation.

We draw the shape in below.

How to Rotate a Shape Without Using Tracing Paper

To rotate a shape without tracing paper, draw horizontal and vertical arrows from the centre of rotation to each corner of the shape. The new corner can be found by rotating each of these arrows according to the following rules:

For example, rotate the shape 90° clockwise without using tracing paper.

The first step is to draw horizontal and vertical arrows connecting the centre of rotation to a corner on the shape.

The corner selected below is one square right and one square up from the centre of rotation.

Using the rules above, a right-facing arrow will be facing down following a 90° clockwise rotation. An upwards-facing arrow will be facing right after a 90° clockwise rotation.

Instead of the corner being one right and one up, it will now be one down and one right.

Using the new position of this corner, the rest of the shape can be drawn in. The same process can be repeated for all corners to check the result.

Here is another example. Rotate the shape 270° clockwise without using tracing paper.

Draw horizontal and vertical arrows from the centre of rotation to each corner.

After a 270° clockwise rotation all upwards-facing arrows will be facing left and all left-facing arrows will be facing down.

Once all corners are drawn in their new positions, the rotated shape can be drawn by connecting these together.