Definition, Formula & Examples

Two geometric figures appear similar if their corresponding angles are equal and their corresponding sides are proportional. One ratio is a fraction that compares two quantities. This picture shows two similar figures with 4 sides.

The blue and red figures are similar.

The blue and red figures are similar

The blue figure in this picture is similar to the red figure. Each side of the red figure is twice as large as the corresponding side of the blue figure.

In two similar geometric figures, the ratio of their corresponding sides is called the scale factor. In order to find the scale factor, identify two corresponding sides, one on each figure. Write down the report of one length to the other to find the scale factor from one figure to the other. In this example, the scaling factor from blue to red figure is 1.6 : 3.2, or 1 : 2. It means that for one unit of length on the blue figure, there are two units of length on the red figure. The scale factor from the red figure to the blue figure is 3.2 : 1.6, or 2 : 1.

Using the scale factor

If two figures are similar, then different characteristics of the figure can be related using the scale factor. For example, think of two squares that are similar. One has a lateral length of 2 inches and the other has a lateral length of 4 inches. That gives a scale factor of 1 : 2 from the small square to the large square.

These two similar squares have a scale factor of 1 : 2 from the small square to the large square.

Similar squares

In order to obtain the length of the side of one square given the length of the side of the other, you can multiply or divide by the scale factor. Let’s see this with the squares shown above.

Suppose you are told that the smallest square has a side length of 2 inches and the scale factor from the smallest to the largest is 1 : 2. Remember, this means that 1 inch on the smallest square is 2 inches on the largest square. If we multiply by the scale factor, 1/2, we will get a smaller number. Then we have to ‘divide’ by the scale factor to get a larger number. To obtain the perimeter of one square given the perimeter of the other, we can multiply or divide by the scale factor. The smallest square has a perimeter of 8 inches. We want to find the perimeter of the largest square. Once again we will have to divide by the scale factor of 1 : 2.

The larger square will have a 16-inch perimeter. Does that make sense? Ask yourself, am I going from a smaller to a larger figure, or from a larger to a smaller one? In this case, we’ve gone from a smaller figure to a larger one, so we expect our answer to be larger than the original perimeter.

To get the area of one square given the area of the other, you can multiply or divide by the square of the scale factor. In our example, the smallest square has an area of 4 square inches. Just as we divided by the scale factor to determine the perimeter of the largest square, we will now divide by the square of the scale factor.

To obtain the volume of one cube given the volume of the other, you can multiply or divide by the cube of the scale factor. To decide whether to multiply or divide, one must consider the value of the scale factor and whether one goes from a larger to a smaller figure, or from a smaller to a larger one.

Imagine that instead of squares, we have two cubes, like big dice. The smaller cube has a side length of 2 inches, and the other has a side length of 6 inches. The volume of the first cube is 8 cubic inches, and the scale factor from the largest cube to the smallest is 3 : 1. To find the volume of the largest cube, we multiply the cube by the scale factor (known as our scale factor is larger than one; we know we want a larger answer, so we multiply rather than divide).

The volume of the second cube is therefore 216 cubic inches. If we check our work, we can see it:

We can use these examples to find the side length, perimeter, area and volume for any pair of similar figures. This makes knowledge of the scale factor incredibly useful.