14.1: Volume Matching (5 minutes)
In this warm-up, students apply the pyramid volume formula to explore how different sets of dimensions in solids can produce the same volume.
Here is a pyramid.
Which, if either, of these solids has the same volume as the pyramid?
Ask students to explain why the rectangular prism has the same volume as the pyramid. The prism has the same base as the pyramid, but it has \(\frac13\) the height. So, when we use \(Bh\) for the prism and \(\frac13 Bh\) for the pyramid, the results are the same.
14.2: Practice with Pyramids (20 minutes)
The purpose of this activity is for students to practice solving problems that involve volumes of pyramids and cones. Monitor for various problem-solving strategies—in particular, for the problem in which students find the radius or height of a cone given a particular volume. Some students may substitute values into a formula and solve the resulting equation, and others may follow a process of deductive reasoning.
Provide students access to scientific calculators. Arrange students in groups of 2. Ask students to discuss their thinking with their partner, and if they disagree, work to reach an agreement.
Supports accessibility for: Memory; Conceptual processing
- Calculate the volume of each solid. Round your answers to the nearest tenth if necessary.
- A particular cone has radius \(r\) and height \(h\).
- Write an expression for the volume of this cone in terms of \(r\) and \(h\).
- What is the height of a cone whose volume is \(16\pi\) cubic units and whose radius is 3 units?
- What is the radius of a cone whose volume is \(16\pi\) cubic units and whose height is 3 units?
Are you ready for more?
The Pyramid of Giza is 455 feet tall. The base is square with a 756-foot side length. How many Olympic-size swimming pool volumes of water can fit inside the Pyramid of Giza? Explain or show your reasoning.
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For the pyramid with a triangular base, students may forget that the triangle area formula involves dividing by 2, resulting in a base area of 12 square units. Prompt them to draw and label the base prior to calculating its area.
Ask students to share their strategies for volume calculations. For example, for the cone, did they calculate the area of the base first, and substitute it into the formula \(V=\frac13 Bh\)? Or, perhaps they wrote out \(V=\frac13 \pi r^2 h\) and performed the calculations all at once.
Select previously identified students to share their thought process on the “working backward” cone problem. If possible, find one student who solved an equation algebraically and another who used backward reasoning rather than formally solving an equation.
Design Principle(s): Cultivate conversation
14.3: An Icy Pyramid (10 minutes)
Students apply their understanding of pyramid volumes to find possible dimensions of a pyramid base given its volume and height.
Monitor for students who start by using trial and error to find values, and for those who begin by finding the area of the base that satisfies the conditions by either substituting the given volume and height into the volume formula and solving for the area of the base, or by using logical reasoning to find the area.
Many students may use squares and rectangles for their bases. Listen for students who consider other shapes such as triangles.
Students should begin the activity without the help of the applet. Once they have found one base that works, they can choose to experiment with the applet to find a second base that works. They will learn to construct a pyramid and find its volume in the embedded applet. For students struggling to draw a base using the Polygon tool, suggest that they select the vertices in counter-clockwise order, and make sure to click again on the first vertex to “close” the polygon. Selecting the vertices in clockwise order will cause the pyramid to extrude downward.
Design Principle(s): Support sense-making
A caterer is making an ice sculpture in the shape of a pyramid for a party. The caterer wants to use 12 liters of water, which is about 720 cubic inches. The sculpture must be 15 inches tall. The caterer needs to decide how large to make the base, which can be any shape.
- Draw and label the dimensions of a base that would work.
- Find a second base that satisfies the baker’s requirements. You may use the applet to help, if you choose.
- Draw your base on the grid with the Polygon tool.
- Change to the 3D Graphics View by clicking on button.
- Click in the 3D window to switch back to the 3D menu.
- Select the Extrude to Pyramid tool and click on your polygon.
- When the dialog box opens, input the height.
- Use the Volume tool to verify your calculations and your figure.
- Refresh the page and repeat the steps with another base that works.
Design Principle(s): Support sense-making
A caterer is making an ice sculpture in the shape of a pyramid for a party. The caterer wants to use 12 liters of water, which is about 720 cubic inches. The sculpture must be 15 inches tall. The caterer needs to decide how large to make the base, which can be any shape. Draw and label the dimensions of 2 different bases that would work.
Students may forget that the pyramid volume formula includes a factor of \(\frac13\). This would lead them to find bases with area 48 square inches. Ask these students to work backward from the dimensions they found to verify if the volume of the pyramid is 720 cubic inches.
Ask previously selected students to share their strategies. Here are some questions for discussion:
- Once you found one set of dimensions, how did you go about working on the second set?
- What did the area of the base need to be, and how did you figure that out?
- Could the base be a circle? If so, how could we find the dimensions of the circle that would work?
Supports accessibility for: Language; Social-emotional skills
The goal of the discussion is to share strategies for working with pyramid volumes. Ask students:
- “How are the expressions \(\frac13 Bh\) and \(\frac13 \pi r^2 h\) related?” (They both give the volume of a cone.)
- “How do you find the volume of an oblique cone or pyramid?” (It’s the same process as for a right cone or pyramid. You just have to make sure to use the actual height of the solid and not the length of one of the edges.)
- “Suppose you know the volume of a pyramid with a rectangular base is 200 cubic units. How could you find a set of dimensions for the pyramid?" (One way is to set up an equation that looks like \(200 = \frac13 Bh\). Then, multiply both sides of the equation by 3 to get \(600 = Bh\). Now you can choose convenient numbers that divide evenly into 600 for the height and the dimensions of the base.)
14.4: Cool-down - Pyramid Dimensions (5 minutes)
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Student Lesson Summary
We can work backward from a given volume to find possible dimensions for a cone or pyramid.
Suppose we want to find dimensions for a cone so it has a volume of \(900\pi\) cubic inches. Start by substituting the volume into the pyramid volume formula to get \(900\pi = \frac13 Bh\). The base of a cone is a circle, so we can write \(900\pi = \frac13 \pi r^2 h\). Multiply both sides of the equation by 3 and divide both sides by \(\pi\) to get \(2,\!700 = r^2 h\).
Now consider different possible values for \(r\) and \(h\). If we can find a perfect square that divides evenly into 2,700, we can set the square root of that number to be the radius. The number 25 is a perfect square and divdes into 2,700, so choose \(r=5\). Now \(2,\!700 = 25h\). This tells us that if the pyramid’s radius is 5 inches, its height is 108 inches because \(2,\!700 \div 25 = 108\).
These aren’t the only possible values. Suppose we set the radius to be 20 inches. Substitute this into the original equation and rearrange to find the value of \(h\).
\(900\pi = \frac13 \pi (20)^2 h\)
\(900\pi = \frac13 \pi \boldcdot 400 h\)
\(2,\!700 = 400 h\)
A height of 6.75 inches together with a radius of 20 inches gives the cone a volume of \(900\pi\) cubic inches.