Learn on PengiReveal Math, AcceleratedUnit 12: Area, Surface Area, and Volume

Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones

In Grade 7 Reveal Math Accelerated, Lesson 12-7 teaches students how to apply the volume formulas V = πr²h for cylinders and V = ⅓πr²h for cones to solve real-world problems. Students explore the relationship between the two formulas, discovering that the volume of a cone is one-third the volume of a cylinder with the same base and height. Practice problems involve calculating volumes using given radius, diameter, and height measurements and finding missing dimensions from known volumes.

Section 1

Volume of a Cylinder

Property

Volume is the amount of space contained within a three-dimensional object. It is measured in cubic units, such as cubic feet or cubic centimeters.

Cylinder Volume Formula:

Section 2

Volume of a Cone

Property

The volume $V$ of a cone with radius $r$ and height $h$ is given by the formula:

V = \frac{1}{3}\pi r^2 h

Examples

A cone with a radius of $3$ cm and a height of $5$ cm has a volume of $V = \frac{1}{3}\pi (3^2)(5) = 15\pi$ cm $^3$ .
A cone with a radius of $4$ inches and a height of $9$ inches has a volume of $V = \frac{1}{3}\pi (4^2)(9) = 48\pi$ inches $^3$ .

Explanation

The volume of a cone measures the amount of space it occupies. This formula shows that the volume depends on the radius of its circular base ( $r$ ) and its perpendicular height ( $h$ ). An important relationship to note is that a cone''s volume is exactly one-third the volume of a cylinder with the same radius and height. To calculate the volume, substitute the known values for the radius and height into the formula.

Section 3

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: $V_{cone} = \frac{1}{3} \cdot V_{cylinder}$

If cylinder volume is $V_{cylinder} = \pi r^2 h$ , then cone volume is $V_{cone} = \frac{1}{3}\pi r^2 h$

Section 4

Solving for a Missing Dimension

Property

To find a missing dimension of a cylinder, rearrange the volume formula $V = \pi r^2 h$ .

To find the height ( $h$ ):
$h = \frac{V}{\pi r^2}$
To find the radius ( $r$ ):
$r = \sqrt{\frac{V}{\pi h}}$

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Volume of a Cylinder

Property

Volume is the amount of space contained within a three-dimensional object. It is measured in cubic units, such as cubic feet or cubic centimeters.

Cylinder Volume Formula:

Section 2

Volume of a Cone

Property

The volume $V$ of a cone with radius $r$ and height $h$ is given by the formula:

V = \frac{1}{3}\pi r^2 h

Examples

A cone with a radius of $3$ cm and a height of $5$ cm has a volume of $V = \frac{1}{3}\pi (3^2)(5) = 15\pi$ cm $^3$ .
A cone with a radius of $4$ inches and a height of $9$ inches has a volume of $V = \frac{1}{3}\pi (4^2)(9) = 48\pi$ inches $^3$ .

Explanation

Section 3

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: $V_{cone} = \frac{1}{3} \cdot V_{cylinder}$

If cylinder volume is $V_{cylinder} = \pi r^2 h$ , then cone volume is $V_{cone} = \frac{1}{3}\pi r^2 h$

Section 4

Solving for a Missing Dimension

Property

To find a missing dimension of a cylinder, rearrange the volume formula $V = \pi r^2 h$ .

To find the height ( $h$ ):
$h = \frac{V}{\pi r^2}$
To find the radius ( $r$ ):
$r = \sqrt{\frac{V}{\pi h}}$

Examples

Overview

Lesson overview

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

Overview

Section 1

Volume of a Cylinder

Property

Volume is the amount of space contained within a three-dimensional object. It is measured in cubic units, such as cubic feet or cubic centimeters.

Cylinder Volume Formula:

Section 2

Volume of a Cone

Property

The volume $V$ of a cone with radius $r$ and height $h$ is given by the formula:

V = \frac{1}{3}\pi r^2 h

Examples

A cone with a radius of $3$ cm and a height of $5$ cm has a volume of $V = \frac{1}{3}\pi (3^2)(5) = 15\pi$ cm $^3$ .
A cone with a radius of $4$ inches and a height of $9$ inches has a volume of $V = \frac{1}{3}\pi (4^2)(9) = 48\pi$ inches $^3$ .

Explanation

Section 3

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: $V_{cone} = \frac{1}{3} \cdot V_{cylinder}$

If cylinder volume is $V_{cylinder} = \pi r^2 h$ , then cone volume is $V_{cone} = \frac{1}{3}\pi r^2 h$

Section 4

Solving for a Missing Dimension

Property

To find a missing dimension of a cylinder, rearrange the volume formula $V = \pi r^2 h$ .

To find the height ( $h$ ):
$h = \frac{V}{\pi r^2}$
To find the radius ( $r$ ):
$r = \sqrt{\frac{V}{\pi h}}$