When a growing soccer player moves from one size soccer ball to another, the ball is re-sized based on the level of the player. The older the player, the larger the soccer ball. Note that the shape of the ball never changes—just the size. The idea of transforming a shape by changing its size is key to the geometry concept of similarity. Changing a shape's size is the type of transformation that results in a similar shape.
Look closely at the image above. What can you say about the rate at which the ball size is increased?
Each ball is exactly two inches bigger around that the previous ball. In other words, these soccer balls are re-sized by a steady factor of two inches. The rate at which an object or shape increases in size is called a scale factor. A scale factor is found by taking the ratio of a length, in this case the circumference, as the shape gets larger or smaller. In this case, the scale factor is \(\small\mathsf{ \frac{26}{24} }\), which is approximately 1.08. Each consecutive ball size is increased by a scale factor of 1.08.
Question
As you grew, you may have gone from riding a bike with 16-inch tires to riding a bike with 20-inch tires. What scale factor is used to increase the size of bike wheels for young riders?
