The scale factor

When shapes are similar, their areas and volumes are related by a ratio. The scale factor is denoted as k and can be obtained through different means, depending on the similar shape.

k can be found by dividing the radii of one circle with another similar one. In the case of similar rectangles, the length of one rectangle is divided by the corresponding length of the other. And finally, in similar triangles, the height or length of one triangle can be divided by the corresponding height or length of the other, to obtain k.

Scale factors and areas

If two shapes are found to be similar, their areas are related by a factor k^2.

Similar circles

similar circles

 

The scale factor from the small circle to the bigger circle is k = \frac{5}{3}
Therefore, the area of the larger circle is  (\frac{5}{3})^2 times the area of the smaller circle. On the contrary, the scale factor of the larger circle to the smaller circle could be
k = \frac{3}{5}. Which means, given the area of the larger circle, to find the smaller one, multiple the area by k = (\frac{3}{5})^2

Similar rectangles

similar rectangles

 

The scale factor from the small rectangle to the large one is k = \frac{8}{6} or \frac{4}{3}. This means the area of the large rectangle is k = (\frac{4}{3})^2 times bigger than the smaller one.

Similar triangles

similar trianglesIn the same way, k from the smaller triangle to the larger one is 3. k from the larger triangle to the smaller one is \frac{1}{3}.

Scale factors and volumes

The volume of two similar shapes are related by the ratio of k^3. This follows the same concept as that involving the scale factor and the area. However, the relationship between the volume of the similar shapes is the cube of the scale factor, k.

A simple note to make is that, if the shape is getting larger, the area and volumes should be multiplied by a number larger than 1. This is a simple check in order to ensure that the right factor is being used.

See also: