Learn on PengiBig Ideas Math, Course 3Chapter 8: Volume and Similar Solids

Lesson 2: Volumes of Cones

In this Grade 8 lesson from Big Ideas Math Course 3, students learn how to calculate the volume of a cone using the formula V = ⅓Bh, discovering that a cone holds exactly one-third the volume of a cylinder with the same base and height. Students apply the formula to find both the volume and the height of right and oblique cones, including real-life problem solving. The lesson aligns with Common Core standard 8.G.9 and builds on prior knowledge of cylinder and pyramid volume formulas.

Section 1

Volume of a Cone Using Base Area

Property

The volume $V$ of a cone is one-third the product of its base area $B$ and its perpendicular height $h$ .

V = \frac{1}{3}Bh

Section 2

Volume of a Cone

Property

A cone is a three-dimensional shape with a circular base and an apex. The volume of a right circular cone is one-third the product of the area of the base and the height. If the height is $h$ and the radius of the base is $r$ , then $V = \frac{1}{3} \pi r^2 h$ .

Examples

A party hat is a cone with a height of 8 inches and a base radius of 3 inches. Its volume is $V = \frac{1}{3} \pi (3^2)(8) = 24\pi$ cubic inches.
A cylinder has a volume of $90$ cubic cm. A cone with the same base and height would have a volume of $V = \frac{1}{3} (90) = 30$ cubic cm.
An hourglass cone holds $15\pi$ cubic inches of sand and has a base radius of 3 inches. Its height is found by solving $15\pi = \frac{1}{3} \pi (3^2)h$ , which gives $h=5$ inches.

Explanation

A cone's volume is directly related to a cylinder's. For a cone and cylinder with the same base radius and height, the cone's volume is exactly one-third of the cylinder's. You could fit three full cones of water into one cylinder.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Volume of a Cone Using Base Area

Property

The volume $V$ of a cone is one-third the product of its base area $B$ and its perpendicular height $h$ .

V = \frac{1}{3}Bh

Section 2

Volume of a Cone

Property

Examples

A party hat is a cone with a height of 8 inches and a base radius of 3 inches. Its volume is $V = \frac{1}{3} \pi (3^2)(8) = 24\pi$ cubic inches.
A cylinder has a volume of $90$ cubic cm. A cone with the same base and height would have a volume of $V = \frac{1}{3} (90) = 30$ cubic cm.
An hourglass cone holds $15\pi$ cubic inches of sand and has a base radius of 3 inches. Its height is found by solving $15\pi = \frac{1}{3} \pi (3^2)h$ , which gives $h=5$ inches.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Volume of a Cone Using Base Area

Property

The volume $V$ of a cone is one-third the product of its base area $B$ and its perpendicular height $h$ .

V = \frac{1}{3}Bh

Section 2

Volume of a Cone

Property

Examples

A party hat is a cone with a height of 8 inches and a base radius of 3 inches. Its volume is $V = \frac{1}{3} \pi (3^2)(8) = 24\pi$ cubic inches.
A cylinder has a volume of $90$ cubic cm. A cone with the same base and height would have a volume of $V = \frac{1}{3} (90) = 30$ cubic cm.
An hourglass cone holds $15\pi$ cubic inches of sand and has a base radius of 3 inches. Its height is found by solving $15\pi = \frac{1}{3} \pi (3^2)h$ , which gives $h=5$ inches.