# Lesson 19

The Volume of a Cone

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

- Calculate the volume of a cone and cylinder given the height and radius, and explain (orally) the solution method.
- Compare the volumes of a cone and a cylinder with the same base and height, and explain (orally and in writing) the relationship between the volumes.

### Student Facing

Let’s explore cones and their volumes.

### Required Preparation

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.

### Learning Targets

### Student Facing

- I can find the volume of a cone in mathematical and real-world situations.
- I know the formula for the volume of a cone.

### CCSS Standards

Addressing

### Print Formatted Materials

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Student Task Statements | docx | |

Cumulative Practice Problem Set | docx | |

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Teacher Presentation Materials | docx |