This lesson is optional since it contains work beyond the scope of the grade. The main purpose of this lesson is to understand that if two of the dimensions of a three-dimensional figure are scaled by the same factor, the volume scales by the square of that factor. A secondary purpose is to see some interesting examples of nonlinear functions arising from geometry.

Students begin the lesson by considering rectangular prisms with a square base and a fixed height. They see that if the side length of the square base of the prism is tripled, the volume is multiplied by 9, which is . The following activity invites students to explore how the volume changes as a function of the radius for cones with a fixed height. Using the structure of the formula for the volume of a cone, they see that tripling the radius leads to a similar result as with the prism—the volume scales by a factor of (MP2).