Curriculum  /  Math  /  10th Grade  /  Unit 6: Three-Dimensional Measurement and Application  /  Lesson 10

Three-Dimensional Measurement and Application

Lesson 10

Math

Unit 6

10th Grade

Lesson 10 of 18

Objective

Describe Cavalieri’s principle relating equal area cross-sections and volume, and how this relates to the formulas for volume. Derive the volume of a sphere using Cavalieri’s principle.

Common Core Standards

Core Standards

  • G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
  • G.GMD.A.2 — Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Foundational Standards

  • 7.G.A.3

Criteria for Success

  1. Describe Cavalieri’s principle as stating if two solids that lie between parallel planes have equal heights and all cross-sections at equal distances from their bases have equal areas, then the solids have equal volumes. 
  2. Relate the general formula for a cylinder to Cavalieri’s principle by describing that if you stack up all cross-sections of the base you will have the volume (multiply by the height).
  3. Describe that even if two solids are not the same shape, you can have equal volume by using Cavalieri’s principle. 

Tips for Teachers

  • The following YouTube video may be helpful to understand Cavalieri’s principle more generally and the derivation of the formula for the volume of a sphere: “Sphere Volume Proof (Math? Help! #39) #KhanAcademyTalentSearch,” by John Dixon.
  • The following GeoGebra applet may be helpful to demonstrate Cavalieri’s principle, which can be done after Anchor Problem #1: GeoGebra, “Cavalieri’s Principal,” by Anthony C.M OR.
  • After Anchor Problem #1, instead of the GeoGebra applet to demonstrate Cavalieri’s principle, you could use a slinky or a stack of pennies to model an oblique and right cylinder. Use "Cavalieri’s Principle,” by Jennifer Wilson on her blog Easing the Hurry Syndrome.
  • Anchor Problem #2 will require the teacher to do a significant amount of modeling to derive the volume of a sphere using Cavalieri’s principle.
  • This lesson focuses on understanding Cavalieri’s principle to derive the volume formula for a sphere as conceptual understanding. Students will receive practice and procedural fluency with finding the volume of a sphere in Lesson 12. 
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Anchor Problems

25-30 minutes

Problem 1

The bases of triangular prism $$T$$ and rectangular prism $$R$$ below lie in the same plane. A plane that is parallel to the bases and also a distance of 3 units from the bottom base intersects both solids and creates cross-sections $$T'$$ and $$R'$$, respectively.

  1.  Find the area of the cross-section $$T'$$ and $$R'$$.
  2. Find the volume of triangular prism $$T$$ and rectangular prism $$R$$.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 3 > Topic B > Lesson 10 — Opening Exercise

Geometry > Module 3 > Topic B > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Below are two aquariums.

In Aquarium A, the cylinder is completely filled with water except for a cone that juts out from the floor and extends to the same height as the cylinder. The cone has the same base area as the cylinder.

In Aquarium B, the shape is half of a 10-foot diameter sphere. The aquarium is mounted to the ceiling of the room.

  1. How many cubic feet of water will Aquarium A hold?
  2. For each aquarium, what is the area of the water’s surface when filled to the maximum height?  
  3. Using parts (a) and (b) and your knowledge of Cavalieri’s principle, find the volume of Aquarium B. 

Guiding Questions

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References

Illustrative Mathematics Use Cavalieri's Principle to Compare Aquarium Volumes

Use Cavalieri's Principle to Compare Aquarium Volumes, accessed on June 1, 2017, 7:40 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Target Task

5-10 minutes

Problem 1

Morgan tells you that Cavalieri’s principle cannot apply to the three-dimensional solids shown below because their bases are different shapes. Do you agree or disagree? Explain your reasoning.

References

EngageNY Mathematics Geometry > Module 3 > Topic B > Lesson 10 — Exit Ticket, Question #1

Geometry > Module 3 > Topic B > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

A triangular prism has an isosceles right triangular base with a hypotenuse of $$\sqrt{32}$$ and a prism height of 15. A square prism has a height of 15, and its volume is equal to that of the triangular prism. What are the dimensions of the square base?

References

EngageNY Mathematics Geometry > Module 3 > Topic B > Lesson 10 — Exit Ticket, Question #2

Geometry > Module 3 > Topic B > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Next

Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.

Lesson 11
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Area and Circumference of Circles

Topic B: Three-Dimensional Concepts and General Volume

Topic C: Cavalieri's Principle, Spheres, and Composite Volume

Topic D: Surface Area, Scaling, and Modeling with Geometry

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