Lesson 13
Building a Volume Formula for a Pyramid
13.1: Cover Your Bases (5 minutes)
Warm-up
In this warm-up, students make connections between a pyramid or cone and a prism or cylinder with a congruent base and equal height. This will be helpful as students work with the volume formula for pyramids and cones in upcoming activities.
Monitor for students who draw an oblique cylinder and for those who draw a right cylinder.
Student Facing
Two solids are shown.
A
B
For each solid, draw and label a prism or cylinder that has a base congruent to the solid’s and a height equal to the solid’s.
Student Response
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Anticipated Misconceptions
Students may not be sure if the cylinder must be oblique or right. Point out that the directions aren’t specific, and they can choose which they prefer to draw.
Activity Synthesis
Invite previously identified students to share their cylinder drawings. If no students drew an oblique cylinder, sketch one for them and ask whether it meets the requirements of the task. Ask students how the volumes of a right cylinder and an oblique cylinder from this activity would compare. As students previously saw with a stack of coins that was shifted to look like an oblique cylinder, the volumes of the two cylinders are the same.
13.2: Splitting a Prism into Pyramids (15 minutes)
Activity
In this task, students derive a formula for the volume of a triangular pyramid by visualizing the split of a triangular prism into 3 pyramids of equal volume. The arguments in this activity are meant to be informal. It’s not necessary, for example, that students formally prove that cutting a rectangle down its diagonal produces two congruent triangles.
As students compare pairs of pyramids, draw conclusions about their volumes, and extend the ideas to include all triangular pyramids, they are making sense of a problem and persevering to solve it (MP1).
Launch
Arrange students in groups of 3. Make sure they have their assembled pyramids from a previous activity.
Design Principle(s): Maximize meta-awareness; Support sense-making
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Here is a triangular prism.
Suppose we split the prism into pyramids like the ones you built earlier. The first pyramid is split off by slicing through points \(E\), \(D\), and \(C\). The remaining part of the prism is sliced through \(B\), \(C\), and \(D\).
P1
P2
P3
- Using the pyramids you built, compare pyramids P1 and P3.
- Think of the faces marked P1 and P3 as the bases of the pyramids. These triangles are the two bases of the original prism. How do the areas of these two bases compare?
- How do the heights of pyramids P1 and P3 compare? Explain your reasoning.
- How do the volumes of pyramids P1 and P3 compare? Explain your reasoning.
- Using the pyramids you built, compare pyramids P2 and P3.
- Think of the gray shaded triangles as the bases of the pyramids. These are formed by slicing one of the prism’s rectangular faces down its diagonal. How do the areas of these two bases compare?
- The heights of pyramids P2 and P3 are equal because when assembled into the prism, the height lines coincide along the length of the prism. How, then, do the volumes of these pyramids compare? Explain your reasoning.
- Based on your answers, how does the volume of each pyramid compare to the volume of the prism?
- How could you use this information to find the volume of one of the pyramids?
Student Response
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Anticipated Misconceptions
Students may not be convinced that the pairs of pyramids have the same volumes. Remind them of the stack of coins activity, and the work they did in previous lessons with oblique figures. If many groups are struggling, consider coming together for a whole-class discussion.
Activity Synthesis
The goal of the discussion is to help students extend these ideas to all triangular pyramids, not just the specific ones used in the lesson. Here are some questions for discussion:
- “Do any of these arguments depend on the specific triangular prism we used?” (No. If you sliced any triangular prism apart into three pyramids like this, you’d still have congruent bases and equal heights for pyramid pairs P2/P3 and P1/P3.)
Display this image for all to see.
- “This pyramid doesn’t look like the ones we used in the lesson. Is it possible to create 3 pyramids with the same volume as this one that could be assembled into a prism?” (The first step would be to shift the apex of the pyramid so that it’s directly over one vertex, resembling the P1 pyramid in earlier activities. This step does not change the volume of the pyramid. Then the pyramid can be duplicated, flipped upside down, and shifted to resemble the P3 pyramid. Finally, the pyramid could be duplicated and shifted to look like P2.)
Display this applet for all to see, moving the slider to show a similar process for a square prism.
- “How, then, could we find the volume of this pyramid?” (Just like in the activity, we could find the volume of the prism with congruent base and equal height, then multiply by \(\frac13\).)
- “Could we use a formula to find the volume? What would it look like?” (Yes. The formula could be written \(V=\frac13 Bh\) or \(V=\frac{Bh}{3}\).)
13.3: Comparing Cross Sections (15 minutes)
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.
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 \(k=0.5\).
- 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 \(\ell\).
Student Response
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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.)
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Lesson Synthesis
Lesson Synthesis
Ask students to summarize the similarities and differences between prisms, cylinders, cones, and pyramids. For prisms and cylinders, if we take cross sections parallel to their bases, all the cross sections are congruent. For pyramids and cones, cross sections taken parallel to the base are similar to each other, not congruent. For a prism or cylinder whose base has area \(B\) and height \(h\), the volume is given by \(Bh\). For a pyramid or cone with the same height \(h\) and whose base also has an area of \(B\) square units, the volume is given by \(\frac13 Bh\).
Invite students to categorize cones and cylinders. Do students think they should be characterized as types of pyramids and prisms, or are they their own entities? There is no one correct answer to this question.
Ask students to add this theorem to their reference charts as you add it to the class reference chart:
A pyramid or cone whose base has area \(B\) square units and whose height is \(h\) units has volume \(\frac13 Bh\) cubic units, regardless of the shape of the base or whether the solid is oblique. (Theorem)
13.4: Cool-down - Pyramid Strategies (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
In an earlier activity, we conjectured that a triangular pyramid has one-third the volume of the prism that has the same height and a base congruent to the pyramid’s. This turns out to be true.
Suppose we have a pyramid that doesn’t have a triangular base. Call the area of its base \(B\) and its height \(h\). To understand how to find the volume of such a solid, think about a triangular pyramid that also has an area of \(B\) square units and a height of \(h\).
Now find a cross section of each solid by dilating the solid’s base using the apex as a center with some scale factor \(k\) between 0 and 1. The area of the cross section in both solids will be \(Bk^2\). This is true for any value of \(k\). Since the cross sections at all heights have equal area, the solids have the same volume. The same idea would apply to a cone. This means the expression \(\frac13 Bh\) gives the volume of any pyramid or cone.
For example, this pyramid’s base has an area of 100 square units. The volume of the pyramid is about 233.3 cubic units, because \(\frac13 \boldcdot 100 \boldcdot 7 \approx 233.3\).
