The Volume of a Cone
5 min
Teacher Prep
Setup
Required Preparation
If possible, provide students access to geometric solids with the same base and height measurements.
Narrative
The purpose of this activity is to get students thinking about how the volume of a cone might relate to the volume of a cylinder with the same base and height (MP1). Additionally, students learn one method for sketching a cone.
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 or 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.
Student Task
The cone and cylinder have the same height, and the radii of their bases are equal.
-
Which figure has a larger volume?
-
Do you think the volume of the cone is more or less than 21 the volume of the cylinder? Explain your reasoning.
- 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.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.
Sample Response
- The cylinder has a larger volume.
- Sample response: I don’t think the cone is more than half the volume because I don’t think I would fit two cones inside the cylinder.
- Answers vary.
Activity Synthesis (Teacher Notes)
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 students' best guesses, and tell them that we will find out the actual fractional amount in the next activity.
If time allows, end the discussion by selecting 2–3 students to share their sketches. Otherwise, display these for all to see and compare the different heights and radii. If no student draws a perpendicular height or slant height, use these images to remind students 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.
Anticipated Misconceptions
If students think the two shapes will have the same volume, ask them to imagine dropping the cone into the cylinder and having extra space around the cone and still inside the cylinder.
Standards
- 8.G.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- 8.G.C.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
The Volume of a Cone
5 min
Teacher Prep
Setup
Required Preparation
If possible, provide students access to geometric solids with the same base and height measurements.
Narrative
The purpose of this activity is to get students thinking about how the volume of a cone might relate to the volume of a cylinder with the same base and height (MP1). Additionally, students learn one method for sketching a cone.
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 or 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.
Student Task
The cone and cylinder have the same height, and the radii of their bases are equal.
-
Which figure has a larger volume?
-
Do you think the volume of the cone is more or less than 21 the volume of the cylinder? Explain your reasoning.
- 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.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.
Sample Response
- The cylinder has a larger volume.
- Sample response: I don’t think the cone is more than half the volume because I don’t think I would fit two cones inside the cylinder.
- Answers vary.
Activity Synthesis (Teacher Notes)
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 students' best guesses, and tell them that we will find out the actual fractional amount in the next activity.
If time allows, end the discussion by selecting 2–3 students to share their sketches. Otherwise, display these for all to see and compare the different heights and radii. If no student draws a perpendicular height or slant height, use these images to remind students 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.
Anticipated Misconceptions
If students think the two shapes will have the same volume, ask them to imagine dropping the cone into the cylinder and having extra space around the cone and still inside the cylinder.
Standards
- 8.G.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- 8.G.C.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.