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 realworld situations.
 I know the formula for the volume of a cone.
CCSS Standards
Addressing
Glossary Entries

base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.

cone
A cone is a threedimensional figure like a pyramid, but the base is a circle.

cross section
A cross section is the new face you see when you slice through a threedimensional figure.
For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

cylinder
A cylinder is a threedimensional figure like a prism, but with bases that are circles.

prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.

pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.

sphere
A sphere is a threedimensional figure in which all crosssections in every direction are circles.

surface area
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm^{2}, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm^{2}.

volume
Volume is the number of cubic units that fill a threedimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units^{3}, because it is composed of 3 layers that are each 20 units^{3}.
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