Lesson 7 Footprints in the Sand Solidify Understanding

Jump Start

Draw diagrams to represent the following shapes:

1.

A right prism with a triangular base.

2.

A right prism with a hexagonal base.

Learning Focus

Derive and use formulas for right prisms and pyramids.

How are the formulas for pyramids and cones related to the formulas for prisms and cylinders?

Can we visualize why this is so?

Open Up the Math: Launch, Explore, Discuss

Benji, Chau, and Kassandra are discussing the various 3-D shapes they plan to include in their sand castles. They are wondering how to calculate the volume of some of the shapes they want to include. Chau wants to include prisms with equilateral triangular bases, and Kassandra wants to include prisms with regular hexagonal bases. Benji only knows that the formula for a rectangular prism is $L \times W \times H$ , and so he is trying to figure out how the shape of the base affects the volume of the prism.

Benji wonders if thinking about the footprint of the prisms Chau and Kassandra want to include in the sand castles will help him figure out their volumes.

Chau wants to include a triangular prism with bases that are equilateral triangles, $2 inches$ on a side and $10 inches$ tall. Benji is examining the footprint of Chau’s prism, inscribed in a rectangle.

1.

Develop a strategy for finding the volume of Chau’s prism, using this drawing that Benji created to help him visualize the footprint of Chau’s triangular prism.

Kassandra wants to include a hexagonal prism with bases that are regular hexagons, $2 inches$ on a side, and the prism is $10 inches$ tall. Benji is examining the footprint of Kassandra’s prism, inscribed in a circle.

2.

Develop a strategy for finding the volume of Kassandra’s prism, using this drawing that Benji created to help him visualize the footprint of Kassandra’s hexagonal prism.

3.

Describe a general procedure for finding the volume of a prism when you are given a description and dimensions of the bases of the prism.

4.

Consider a prism with a height $h$ and a regular $n$ -sided polygon inscribed in a circle of radius $r$ as a base. How can you use this description of a prism to justify why the volume of a cylinder is given by $V = π r^{2} h$ .

Pause and Reflect

Benji has described his strategy for finding the volume of any prism to Chau and Kassandra. They are both excited by his findings, but Kassandra has another question: “I have always wondered why the volume of pyramids or cones is always $\frac{1}{3}$ of the volume of the prism or cylinder with the same base and height.”

Chau replies, “I’m not sure why it is true in general, but I think I can explain it for a square pyramid whose height is $\frac{1}{2}$ of the side length of the square that forms the base.” Chau quickly sketches the following cube with all four of its diagonals. She has labeled the length of each edge of the cube as $x inches$ .

5.

The diagonals divide the cube into $6$ congruent pyramids. (Each face of the cube is the base of one of the pyramids.) How is the volume of each of these pyramids related to the volume of the cube? Use Chau’s drawing and the relationship between the volumes of the cube and the pyramids to derive a formula for the volume of one of the pyramids in terms of $x$ .

6.

The pyramid in problem 5 does not have the same height as the cube. Find the volume of the rectangular prism that has the same base and height as one of the pyramids.

7.

How is the volume of the pyramid described in problem 5 related to the volume of the rectangular prism described in problem 6?

Ready for More?

The diagram shows the footprints for two different pyramids: a pyramid with a rectangular base and a pyramid with an equilateral triangular base. How do the volumes of these two pyramids compare? What work did you do to determine your answer?

Takeaways

Two strategies for finding the volume of a right prism:

Strategy 1: A dissection method

Strategy 2: Using a formula

How to find the volume of a square pyramid:

Vocabulary

pyramid
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we derived a formula for the volume of right prisms with non-rectangular bases, and a formula for the volume of pyramids. These formulas were derived by decomposing rectangular prisms in various ways and considering how the sum of the volumes of the resulting prisms or pyramids had to add up to the volume of the original rectangular prism.

Retrieval

1.

Graph the two points on the coordinate grid. Then find the length of the segment that connects them.

a.

$A (- 4, 4)$ , $B (2, 4)$

b.

$P (- 4, 4)$ , $Q (2, - 4)$

2.

Find the measure of angle $A$ .