MAFS.912.G-GMD.1.2

Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Geometry: Geometric Measurement & Dimension
Cluster: Level 3: Strategic Thinking & Complex Reasoning
Cluster: Explain volume formulas and use them to solve problems. (Geometry - Additional Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved

Related Courses

This benchmark is part of these courses.
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1298310: Advanced Topics in Mathematics (formerly 129830A) (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Lesson Plan

Exploring Cavalieri's Principle:

In this 90 minute lesson/exploration of Cavalieri's Principle using technology (GeoGebra 5.0), students calculate volume of oblique solids and determine if Cavalieri's applies to given scenarios.

Students will perform transformations of a base figure in a 3 dimensional coordinate system to observe the creation of right and oblique solid figures. After these observations, students will create a conjecture about calculating the volume of the oblique solids. Students will use the conjecture to determine situations in which Cavalieri's applies and calculate volumes of oblique solids.

The teacher may choose to split lesson into two 45 minute sessions.

Variable representation of volume formulas V = BH and V = 1/3BH:

  • B = Base Area
  • H = Height of Solid
  • b = length of base figure
  • h = height of base figure

The differentiation is used so that the height of the solid is not confused with the height of the base.

Type: Lesson Plan

Problem-Solving Task

Use Cavalieri’s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Problem-Solving Task

Use Cavalieri’s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Task

Use Cavalieri’s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task