Unit 5, Lesson 13
Building a Volume Formula for a Pyramid
Geometry
13.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
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.
Launch
NoneActivity
NoneTwo 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
Loading...
Building on Student Thinking
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
The goal is to make sure students understand that right and oblique cylinders have the same volume as long as the base and height are the same in each figure. 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.
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 and height , the volume is given by . For a pyramid or cone with the same height and whose base also has an area of square units, the volume is given by .
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.
Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
A pyramid or cone whose base has an area of square units and whose height is units has volume cubic units, regardless of the shape of the base or whether the solid is oblique. (Theorem)
Student Lesson Summary
We can use relationships between pyramids and prisms to build a formula for the volume of a pyramid. The image shows three square pyramids assembled into a cube. All three pyramids have the same volume so their volume must be one-third the volume of the cube.
More generally, a triangular pyramid has one-third the volume of the prism that has the same height and a base congruent to the pyramid’s.
Suppose we have a pyramid that doesn’t have a triangular base. Call the area of its base and its height . To understand how to find the volume of such a solid, think about a triangular pyramid that also has an area of square units and a height of .
Now find a cross-section of each solid by dilating the solid’s base using the apex as a center with some scale factor between 0 and 1. The area of the cross-section in both solids will be . This is true for any value of . Because 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 that the expression gives the volume of any pyramid or cone.