Lesson 15

The Volume of a Cone

15.1: Which Has a Larger Volume?

CCSS Standards

• 8.G.C.9

Warm-up: 5 minutes

The purpose of this activity is for students to think about how the volume of a cone might relate to the volume of a cylinder with the same base and height. Additionally, students learn one method for sketching a cone. In this activity, just elicit students’ best guess about how many cone-contents would fit into the cylinder (or, what fraction of the cylinder’s volume is the cone’s volume). In the next activity, they will watch a demonstration that verifies the actual amount.

Launch

If you have access to appropriate geometric solids that include a cylinder and a cone with congruent bases and equal heights, consider showing these to students, even passing them around for students to hold if time permits.

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time, followed by time to discuss fractional amount with partner. Follow with a whole-class discussion.

Conceptual Processing: Manipulatives. Provide manipulatives (i.e., geometric solids) to aid students who benefit from hands-on activities.

Student Facing

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

1. Which figure has a larger volume?
2. Do you think the volume of the smaller one is more or less than $\frac12$ the volume of the larger one? Explain your reasoning.
1. 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 $h$.

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

Invite students to share their answers to the first two questions. The next activity includes a video that shows that it takes 3 cones to fill a cylinder that has the same base and height as the cone, so it is not necessary that students come to an agreement about the second question, just solicit student’s best guesses, and tell them that we will find out the actual fractional amount in the next activity.

End the discussion by selecting 2–3 students to share their sketches. Display these for all to see and compare the different heights and radii. If no student draws a perpendicular height or slant height, display the image shown here for all to see and remind students that in earlier units we learned that height creates a right angle with something in the figure. In the case of the cones, the height is perpendicular to the circular base.

15.2: From Cylinders to Cones

CCSS Standards

• 8.G.C.9

Activity: 20 minutes

In this activity, students use the relationship that the volume of a cone is $\frac13$ of the volume of a cylinder to calculate the volume of various cones. Students start by watching a video (or demonstration) that shows that it takes the contents of 3 cones to fill the cylinder when they have congruent bases and equal heights. Students use this information to calculate the volume of various cones and cylinders. For the last question, identify students who:

• write the equation as $\frac13 V$ (or $V \div 3$), where $V$ represents the volume of a cylinder with the same base and height as the cone.
• write the equation in terms of $r$ and $h$ ($V=\frac13\pi r^2h$ ).

Launch

How Many Cones Does it Take to Fill a Cylinder with the Same Base and Height? from Open Up Resources on Vimeo https://vimeo.com/196520545.

Either conduct a demonstration or show the video and tell students to write down anything they notice or wonder while watching. Pause for a whole-class discussion. Record what students noticed and wondered for all to see. Ensure that students notice that it takes the contents of 3 cones to fill the cylinder, or alternatively, that the volume of the cone is $\frac13$ the volume of the cylinder. Then, set students to work on the questions in the task, followed by a whole-class discussion.

Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.

Conceptual Processing: Eliminate Barriers. Allow students to use calculators to ensure inclusive participation in the activity.

Student Facing

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

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

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

Select previously identified students to share the volume equation they wrote for the last question. Display examples for all to see and ask “Are these equations the same? How can you know for sure?” (The calculated volume is the same when you use both equations.)

If no student suggests it, connect $\frac13 V$, where $V$ represents the volume of a cylinder with the same base and height as the cone, to the volume of the cone, $\frac13\pi r^2h$. Reinforce that these are equivalent expressions.

Add the formula $V=\frac13\pi r^2h$ and a diagram of a cone to your classroom displays of the formulas being developed in this unit.

15.3: Calculate That Cone

CCSS Standards

• 8.G.C.9

Activity: 10 minutes

The purpose of this activity is for students to calculate the volume of cones given their height and radius. Students are given a cylinder with the same height and radius and use the volume relationship they learned in the previous activity to calculate the volume of the cone. They then calculate the volume of a cone given a height and radius using the newly learned formula for volume of a cone. For the last problem, an image is not provided to give students the opportunity to sketch one if they need it.

Launch

Give students 3–5 minutes of quiet work time followed by a whole-class discussion.

Fine Motor Skills: Eliminate Barriers. Provide an enlarged version of the visual.

Student Facing

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 $\pi$.

2. Here is a cone.
1. What is the area of the base? Express your answer in terms of $\pi$.
2. What is the volume of the cone? Express your answer in terms of $\pi$.
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 $\pi$, and give a numerical answer.

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

For the first problem, Invite students to explain how they calculated the volume of both figures and have them share the different strategies they used. If not mentioned by students bring up these strategies:

• Calculate the volume of the cylinder, then divide volume of cylinder by 3 to get the volume of the cone.
• Calculate the volume of the cylinder, then multiply volume of cylinder by $\frac13$ to get the volume of the cone.
• Calculate the volume of the cone, then multiply volume of cone by 3 to get the volume of the cylinder.

For the third problem, ask students to share any sketches they came up with to help them calculate the answer. Explain to students that sometimes we encounter problems that don’t have a visual example and only a written description. By using sketches to help to visualize what is being described in a problem, we can better understand what is being asked.

15.4: Calculate Volumes of Two Figures

CCSS Standards

• 8.G.C.9

Student Facing

A cone with the same base but a height 3 times taller than the given cylinder exists. What is the volume of each figure? Express your answers in terms of $\pi$.

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Student Lesson Summary

Student Facing

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 $63\pi$ cubic feet since $\pi \boldcdot 3^2 \boldcdot 7 = 63\pi$. The cone has a volume that is $\frac13$ of that, or $21\pi$ cubic feet.

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