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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.
Standard #: MAFS.912.G-GMD.1.1Archived Standard
Standard Information
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
Content Complexity Rating: Level 3: Strategic Thinking & Complex Reasoning - More Information
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes
Related Courses
Related Resources
Formative Assessments
  • Volume of a Cylinder Students are asked to derive and explain a formula for the volume of a cylinder given a prism with the same height and the same cross-sectional area at every height.
  • Area and Circumference – 1 This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.
  • Volume of a Cone Students are asked to derive and explain a formula for the volume of a cone given a pyramid with the same height and the same cross-sectional area at every height.
  • Area and Circumference - 3 This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.
  • Area and Circumference - 2 This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r2 · A(1)].
  • Volume of a Pyramid Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.
Lesson Plans
  • Filled to Capacity! This is a lesson where students investigate, compare, dissect, and use the relationship between volume of a cone and cylinder with equal corresponding dimensions.
  • The Relationship Between Cones and Cylinders Students create a cone and a cylinder with the same height and base. At the conclusion of the lesson, the students will know that the volume ratio between the cone and cylinder is 1:3.
  • Discovering the Formulas for Circumference and Area of a Circle Using reasoning skills, students will understand how the formulas for circumference and area of a circle are derived. Students will use a wide array of skills such as deductive reasoning, finding patterns, using algebra, modeling and transformation of an object. The teacher ensures student success through direct instruction, investigation and collaborative group work.
Perspectives Video: Professional/Enthusiast
Perspectives Video: Teaching Idea
  • Robot Mathematics: Gearing Ratio Calculations for Performance A robotics teacher clearly breaks down the parts of the gear ratio formula, showing how to derive and calculate it. He also shows a method for helping students to visually discover Pi. He rolls a wheel on paper for one rotation to measure the circumference and then measures the number of times the diameter of that wheel fits in that rotation.
MFAS Formative Assessments
  • Area and Circumference – 1 This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.
  • Area and Circumference - 2 This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r2 · A(1)].
  • Area and Circumference - 3 This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.
  • Volume of a Cone Students are asked to derive and explain a formula for the volume of a cone given a pyramid with the same height and the same cross-sectional area at every height.
  • Volume of a Cylinder Students are asked to derive and explain a formula for the volume of a cylinder given a prism with the same height and the same cross-sectional area at every height.
  • Volume of a Pyramid Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.
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