options: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

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.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
problem_activity: Activity
Students generalize the process for finding the volume of a triangular pyramid, concluding it applies to all pyramids. As they compare cross sections across different solids, they are looking for and making use of structure (MP7).
Launch
Tell students that in this activity, they’ll decide whether their method to find the volume of triangular pyramids extends to other kinds of pyramids and cones.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students how to calculate the area of a dilated triangle given the area of the original triangle and a scale factor. Ask students how they can use this method to calculate the area of each of the three cross sections.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Each solid in the image has height 6 units. The area of each solid’s base is 10 square units. A cross section has been created in each by dilating the base using the apex as a center with scale factor
\(k=0.5\)
.
[IMG]
Calculate the area of each of the 3 cross sections.
Suppose a new cross section was created in each solid, all at the same height, using some scale factor
\(k\)
. How would the areas of these 3 cross sections compare? Explain your reasoning.
What does this information about cross sections tell you about the volumes of the 3 solids?
Calculate the volume of each of the solids.
Student Response
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Student Facing
Are you ready for more?
An octahedron is a solid whose faces consist of 8 equilateral triangles. Find the volume of an octahedron with edge length
\(\ell\)
.
[IMG]
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may believe the cross sections have area 5 square units rather than 2.5 square units. Remind them that if a two-dimensional figure is dilated by a factor of
\(k\)
, the area is multiplied by
\(k^2\)
. However, the exact value of the area isn’t as important as the concept that the areas are the same for all 3 cross sections.
Activity Synthesis
In the synthesis discussion, help students use the reasoning developed in the task to extend the formula
\(V=\frac13Bh\)
to all pyramids, not just triangular ones. Ask students:
“How did you calculate the volume of the triangular prism?” (Multiplied the area of the base, 10 square units, by the height, 6 units, then multiplied by
\(\frac13\)
to get 20 cubic units.)
“How did you find the volume of the remaining solids?” (The volumes of all the solids are the same.)
“Suppose we didn’t know the actual values of the height or the area of the base, but we knew all the bases had the same area
\(B\)
and the solids all had the same height
\(h\)
. Would all 3 solids still have the same volume?” (Yes. Each set of cross sections would still have the same areas. Instead of
\(10k^2\)
, each cross section would have area
\(Bk^2\)
.)
“Does the expression
\(\frac13 Bh\)
give the volume for
any
pyramid or cone? Why or why not?” (Yes, it does. For any pyramid or cone, we can draw a triangular pyramid with identical volume, the same height, and a base with equal area. We know that
\(\frac13 Bh\)
works for this triangular pyramid, so it also works for the particular pyramid or cone we’re looking at, too.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their solutions to the last question, present an incorrect answer and explanation. For example, “The volume of the pyramid is 60 cubic units because the volume of a pyramid is the area of the base times the height. The area of the base is 10 square units and the height is 6 units, so the volume is 10 times 6, which is 60.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify that the volume of a pyramid is a third of the volume of a prism with a congruent base and equal height. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to justify why the volume of the pyramid is 20 cubic units. This will help students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 1