Rotations in Math: Definition & Overview

Geometric Transformations

There are four types of geometric transformations:

1. Translation – movement of the object without rotating it or changing its size
2. Reflection – flipping the object about a line of reflection
3. Rotation – rotating a figure about a point
4. Dilation – changing the size of a figure without changing its essential shape

For this lesson, we’ll focus specifically on rotation.

Rotations

A rotation is the movement of a geometric figure about a certain point. The amount of rotation is described in terms of degrees. If the degrees are positive, the rotation is performed counterclockwise; if they are negative, the rotation is clockwise. The figure will not change size or shape, but, unlike a translation, will change direction. The initial figure is always called the pre-image, while the rotated figure will be called the image.

Let’s look at an example. Let’s rotate this figure 90 degrees:

Rotate the figure 90 degrees

When the pre-image is rotated, the final image looks like this, where the maroon-colored line is the pre-image and the blue line is the image:

The figure was rotated 90 degrees

Notation

The mathematical notation for rotation is usually written like this: R (center, rotation), where the center is the point of rotation and the rotation is given in degrees. Often, rotations are written using coordinate notation, which means that their coordinates on the coordinate plane are given. This will help you to draw both the pre-image and the image easily.

Rules of Rotation

There are some general rules for the rotation of objects using the most common degree measures (90 degrees, 180 degrees, and 270 degrees). The general rule for rotation of an object 90 degrees is (x, y) ——–> (-y, x). You can use this rule to rotate a pre-image by taking the points of each vertex, translating them according to the rule, and drawing the image. Take the previous example: the points that mark the ends of the pre-image are (1, 1) and (3, 3). When you rotate the image using the 90 degrees rule, the end points of the image will be (-1, 1) and (-3, 3).

The rules for the other common degree rotations are:

• For 180 degrees, the rule is (x, y) ——–> (-x, –y)
• For 270 degrees, the rule is (x, y) ——–> (y, –x)