Curriculum  /  Math  /  10th Grade  /  Unit 6: Three-Dimensional Measurement and Application  /  Lesson 7

Three-Dimensional Measurement and Application

Lesson 7

Math

Unit 6

10th Grade

Lesson 7 of 18

Objective

Define and calculate the volume of pyramids and cones. Describe the relationship between general cylinders and general cones with the same base area.

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.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Foundational Standards

  • 8.G.C.9

Criteria for Success

  1. Derive the formula for a cone in relation to a cylinder with the same base area and height, and the formula for a pyramid in relation to a prism with the same base area (and shape) and height. 
  2. Describe that pyramids and cones can be right or oblique. In right pyramids and cones, a line drawn from the apex to the center of the base is perpendicular to the base. 
  3. Calculate the volume of pyramids and cones in contextual and non-contextual problems.
  4. Given either the radius, height, or another dimension (like side length of the base of a square pyramid), find other measurements in the problem.

Tips for Teachers

  • Students may need to review finding the volume of pyramids and cones before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill. 
  • Have students use or develop a reference sheet as needed. A good example is the Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet.
  • The following YouTube video is useful for this lesson to discuss the relationship between the volumes of pyramids and prisms: YouTube, “Volume of Three Square Pyramids Fitting into a Cube,” by Lamee Storage.
  • Some of the Problem Set Guidance is from Grade 8 (8.G.9). However, these questions will help the students build procedural fluency calculating the volume of pyramids and cones from their conceptual understanding of the formulas.
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Anchor Problems

25-30 minutes

Problem 1

Find the volume of the right prism and right pyramid below. How do the volumes compare?

Guiding Questions

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Problem 2

A cylindrical glass with a height of 10 centimeters and a radius of 3 centimeters is filled with lemonade. How many cone-shaped cups with half the height of the glass and the same radius can be completely filled with the lemonade from the glass?

Guiding Questions

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Target Task

5-10 minutes

Below is a square pyramid and a square prism. Which has the larger volume? Explain your reasoning.

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 height of the pyramid or cone. 
  • Include questions such as “a cone with a radius of a certain amount is partially filled up to half the height. If the water is poured into a cylinder with the same radius and height as the cone, what fraction of the cylinder will be filled?”

Next

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

Lesson 8
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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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