Three-Dimensional Measurement and Application

Lesson 8

Math

Unit 6

10th Grade

Lesson 8 of 18

Objective


Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms. 

Common Core Standards


Core Standards

  • G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
  • G.GMD.B.4 — Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
  • G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

  • 8.G.B.7

Criteria for Success


  1. Describe how a triangle is formed when a diagonal is drawn across a three-dimensional figure.
  2. Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.
  3. Calculate volume and composite volume of three-dimensional figures using the Pythagorean Theorem to find missing measures.

Tips for Teachers


Some of the Problem Set Guidance is from Grade 8 (8.G.7). However, these questions will help the students build procedural fluency with using the Pythagorean Theorem to calculate the volume of pyramids, prisms, and compound shapes through their conceptual understanding. 

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Anchor Problems


Problem 1

Below is a right rectangular prism. The length of $$\overline{AB}$$ is 4 inches and the length of $$\overline{BC}$$ is 3 inches. The length of $$\overline{MC}$$ is 8 inches. What is the volume of the right rectangular prism?

Guiding Questions

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Problem 2

Robin made a model of a building. The model is composed of a right square pyramid glued exactly onto a right rectangular prism. The model and some of its dimensions are shown in the diagram below.

  1. What is the volume, in cubic inches, of the part of the model that is a right rectangular prism? Show or explain how you got your answer.
  2. What is the height, in inches, of the part of the model that is a right square pyramid? Show or explain how you got your answer. 
  3. What is the volume, in cubic inches, of the entire model? Show or explain how you got your answer.

Guiding Questions

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References

Massachusetts Department of Elementary and Secondary Education Release of November 2013 MCAS Retest ItemsQuestion #31

Release of November 2013 MCAS Retest Items is made available by the Massachusetts Department of Elementary and Secondary Education. © 2017 Commonwealth of Massachusetts. Accessed Sept. 20, 2018, 2:38 p.m..

Modified by Fishtank Learning, Inc.

Target Task


Below is a right circular cylinder with points $$B$$ and $$A$$ that lie on the edge of the cylinder. The length of $$\overline{AB}$$ is 9 centimeters. What is the volume of the cylinder? 

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.

  • Include problems where students must find the diagonal measurement of a three-dimensional object. 
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Lesson 7

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Lesson 9

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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