Lesson 19The Volume of a Cone

Learning Goal

Let’s explore cones and their volumes.

Learning Targets

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

Lesson Terms

cone
cylinder
sphere

Warm Up: Which Has a Larger Volume?

The cone and cylinder have the same height, and the radii of their bases are equal.

A right circular cylinder and a right circular cone. Both the cylinder and the cone have a height of 8.

Which figure has a larger volume?
Do you think the volume of the smaller one is more or less than $\frac{1}{2}$ the volume of the larger one? Explain your reasoning.
Here is a method for quickly sketching a cone:
- Draw an oval.
- Draw a point centered above the oval.
- Connect the edges of the oval to the point.
- Which parts of your drawing would be hidden behind the object? Make these parts dashed lines.
Sketch two different sized cones. The oval doesn’t have to be on the bottom! For each drawing, label the cone’s radius with $r$ and height with $h$ .

Activity 1: From Cylinders to Cones

A right cirular cylinder and a right circular cone. Both the cylinder and the cone have a height labeled h and have a radius labeled r.

A cone and cylinder have the same height and their bases are congruent circles.

If the volume of the cylinder is 90 cm³, what is the volume of the cone?
If the volume of the cone is 120 cm³, what is the volume of the cylinder?
If the volume of the cylinder is $V = π r^{2} h$ , what is the volume of the cone? Either write an expression for the cone or explain the relationship in words.

Activity 2: Calculate That Cone

Problem 1

Here is a cylinder and cone that have the same height and the same base area.

What is the volume of each figure? Express your answers in terms of $π$ .

A right circular cylinder and a right circular cone. The cylinder has a diameter of 10 and a height of 4. In the cone, the height is indicated with dashed line and the radius is also indicated with a dashed line.

Problem 2

Here is a cone.

A right circular cone with a height of 8 and a radius of 6.

What is the area of the base? Express your answer in terms of $π$ .
What is the volume of the cone? Express your answer in terms of $π$ .

Problem 3

A cone-shaped popcorn cup has a radius of 5 centimeters and a height of 9 centimeters. How many cubic centimeters of popcorn can the cup hold? Use 3.14 as an approximation for $π$ , and give a numerical answer.

Are you ready for more?

A grain silo has a cone shaped spout on the bottom in order to regulate the flow of grain out of the silo. The diameter of the silo is 8 feet. The height of the cylindrical part of the silo above the cone spout is 12 feet while the height of the entire silo is 16 feet.

How many cubic feet of grain are held in the cone spout of the silo? How many cubic feet of grain can the entire silo hold?

A cylinder with a cone attached. The cone has a diameter of 8 feet and the cylinder has a height of 12 feet. The enter figure is 16 feet.

Lesson Summary

If a cone and a cylinder have the same base and the same height, then the volume of the cone is $\frac{1}{3}$ of the volume of the cylinder. For example, the cylinder and cone shown here both have a base with radius 3 feet and a height of 7 feet.

The cylinder has a volume of $63 π$ cubic feet since $π \cdot 3^{2} \cdot 7 = 63 π$ . The cone has a volume that is $\frac{1}{3}$ of that, or $21 π$ cubic feet.

An image of a right circular cone and a right circular cylinder. The cone has a height of 7 and radius of 3. The cylinder has a height of 7 and a radius of 3.

If the radius for both is $r$ and the height for both is $h$ , then the volume of the cylinder is $π r^{2} h$ . That means that the volume, $V$ , of the cone is $V = \frac{1}{3} π r^{2} h$