Lesson 19

In this lesson students start working with cones, and learn that the volume of a cone is \(\frac13\) the volume of a cylinder with a congruent base and the same height. First, students learn a method for quickly sketching a cone, and the meaning of the radius and height of a cone. Then they watch a video (or if possible, a live demonstration) showing that it takes three cones of water to fill a cylinder with the same radius and height. At this point, it is taken as a mysterious and beautiful fact that the volume of a cone is one third the volume of the associated cylinder. A proof of this fact requires mathematics beyond grade level.

Students write the volume of a cone given a specific volume of a cylinder with the same base and height, and vice versa. Then they use the formula for the volume of a cylinder learned in previous lessons to write the general formula \(V= \frac13\pi r^2 h\) for the volume, \(V\), of a cone in terms of its height, \(h\), and radius, \(r\). Finally, students practice computing the volumes of some cones. There are opportunities for further practice in the next lesson.

For the Which Has a Larger Volume activity, it is suggested that students have access to geometric solids.

During the From Cylinders to Cones activity, students will need to view a video. Alternatively, do a demonstration with a cone that could be filled with water and poured into a cylinder.