Lesson 15

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

Glossary Entries

  • cone

    A cone is a three-dimensional figure like a pyramid, but the base is a circle.

  • cylinder

    A cylinder is a three-dimensional figure like a prism, but with bases that are circles.

  • sphere

    A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.

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