Lesson 8 Cavalieri to the Rescue Solidify Understanding

Learning Focus

Apply Cavalieri’s principle to 2-D and 3-D figures.

How do you find the volume of “tilted” prisms, cylinders, pyramids, and cones, such as the Leaning Tower of Pisa?

How does Cavalieri’s principle resolve the dilemma of justifying the volume formula for pyramids that do not have square bases, as well as the volume formula for cones?

Where did the formula for the volume of a sphere come from?

Open Up the Math: Launch, Explore, Discuss

Tacen, Jacklyn, and Jacob are playing a geometry game. Each player selects a point $C$ on the line segment $\overset{―}{M N}$ , which is parallel to line segment $\overset{―}{A B}$ . The points $A$ , $B$ , and $C$ form the vertices of a triangle. The player who creates the triangle with the largest area wins the game.

Tacen has placed his point at position $C_{1}$ , Jacklyn has selected point $C_{2}$ , and Jacob has chosen to locate his point at $C_{3}$ . Now they are discussing their choices before calculating the areas of each triangle to determine the winner.

Tacen: I chose my point so the triangle would stretch as far left as possible, enclosing a large amount of area.

Jacklyn: I thought it would be best to create an isosceles triangle so the triangle would be symmetric about its altitude, so I chose a point on segment $M N$ directly above the midpoint of segment $A B$ .

Jacob: I thought that a right triangle would create the largest triangle, since my triangle would be half of a rectangle.

1.

Without doing any calculations, who do you think created the triangle with the largest area?

2.

If you were playing the game with Tacen, Jacklyn, and Jacob, where would you place your point $C$ ? Mark a point on segment $M N$ to represent your best guess. You may mark your point at the same position as one that has already been chosen, if you agree that point would form the largest triangle.

3.

Now it is time to determine a winner. Make any measurements necessary to calculate the winner of the game. Whose strategy won?

Tacen, Jacklyn, and Jacob were initially surprised by the results and wondered why the triangle images were so deceptive. They began to wonder if they could really believe their calculations. Then Tacen suggested an experiment.

He drew a series of line segments in each triangle, with each segment parallel to the base of the triangle, $\overset{―}{A B}$ , and with corresponding segments in each of the triangles drawn at the same distance above the base, as shown in the diagram.

Tacen then measured each of the corresponding line segments.

4.

Complete Tacen’s experiment by measuring each of the corresponding line segments. What do you notice? What does this observation suggest about the areas of the triangles, and why?

Jacklyn said, “It feels like you are treating each triangle as if it was made up of a bunch of layers or slices.”

Jacob, inspired by Jacklyn’s comment, pulled a handful of pennies out of his pocket and stacked them to form a cylinder. “I can calculate the volume of this stack of coins using the formula $V = B h$ . But what if I tilt the stack so it looks more like the Leaning Tower of Pisa? Now how do I figure out how much space the coins occupy?”

Tacen and Jacklyn smiled at Jacob’s clever way of illustrating an idea that was new to both of them. They were excited to tell their geometry teacher about their discovery and Jacob’s principle. They were surprised to hear that Jacob wasn’t the first person to think of it and that it is known as Cavalieri’s principle.

5.

In your own words, state what you think Cavalieri’s principle is, based on the triangle experiment and the stack of coins illustration.

Try out another experiment with Cavalieri’s principle. Once again, line $M N$ is parallel to segment $A B$ . Measure the length of segment $A B$ , and then mark a congruent segment $C D$ anywhere on line $M N$ . Connect the endpoints of segment $A B$ to the endpoints of segment $C D$ to form a parallelogram. Mark another segment $E F$ on line $M N$ so that $E F$ is also congruent to $A B$ . Connect the endpoints of segment $A B$ to the endpoints of segment $E F$ to form another parallelogram.

6.

Use these two noncongruent parallelograms to illustrate Cavalieri’s principle. What can you say about the areas of these two parallelograms, and what convinces you that this is true?

Jacob’s demonstration with the pennies has convinced Tacen and Jacklyn that the volume of prisms and cylinders, where the parallel slices are not directly above each other, is the same as the volume of corresponding right prisms and right circular cylinders with the same base and height. Looking online, they have learned that these types of solids are called oblique prisms, oblique pyramids, oblique cylinders, and oblique cones.

While online, Tacen found this information: If in two solids of equal altitude, the areas of the slices made by planes parallel to and equidistant from their respective bases are always equal, then the volumes of the two solids are equal.

Tacen realizes this means that the corresponding slices only need to have the same area and not the same shape.

7.

In the previous lesson, it was shown that the volume of a pyramid with a square base is given by $V = \frac{1}{3} B h$ , where $B$ is the area of the square base. How can the description of Cavalieri’s principle that Tacen found online be used to argue that the volume of a pyramid with any type of polygon as a base is given by $V = \frac{1}{3} B h$ , where $B$ is the area of the polygon that forms the base?

8.

Consider a cone with an altitude of $h$ and a circle of radius $r$ for a base. How could you use Cavalieri’s principle to prove the volume formula for the cone is $V = \frac{1}{3} π r^{2} h$ ?

9.

Using the following diagram, show that corresponding slices taken at the same distance above the base of the cylinder and the equator of the sphere have the same cross-sectional area. This would imply that the volume of the top half of the sphere is the same as the volume of the cylinder minus the cone. Use this information to derive the volume formula for a sphere.

Ready for More?

Jacklyn and Jacob each find interesting online animations and activities that give them additional insights about Cavalieri’s principle and the volumes of oblique pyramids and cones. Links to each of these resources are given, so you can explore them on your own. Each student has written a brief summary of what the digital resource revealed.

Jacob finds a GeoGebra app that helps him visualize why all right and oblique pyramids with the same base (that is, the bases of the pyramids are congruent shapes) and height will have the same volume: https://openup.org/6pW1S9. He also finds an app that helps him understand the derivation of the volume formula of a sphere: https://www.geogebra.org/m/a9jQQQFz.

1.

Summarize what you learned by exploring Jacob’s apps.

Jacklyn finds a video that help her visualize a way of using the idea of approaching a limit to prove that the volume of a pyramid with a square base is $\frac{1}{3} B h$ .

She is surprised that the mathematics reminds her of the mathematics of Lesson 2, only in a 3-D setting. https://openup.org/H0geF8

2.

Summarize what you learned by exploring Jacklyn’s video.

Takeaways

How to find the volume of a pyramid or cone:

How to find the volume of a sphere:

In your best words, explain Cavalieri’s principle so someone else can understand it. You will want to include an illustration to help support your description.

2-D version:

3-D version:

Vocabulary

Lesson Summary

In this lesson, we learned about Cavalieri’s principle, which can be used as a tool for deriving the volume formulas for oblique prisms, pyramids, cones, and spheres.

Retrieval

1.

Define a kite by answering the questions.

a.

How many sides?

b.

Which sides are $≅$ ?

c.

Which sides are $∥$ ?

d.

How many $∠$ s are $≅$ ?

2.

Determine whether the two solids are similar, congruent, or neither. Justify your answer.

similar

congruent

neither