9.2.4
How much will it hold?
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Volume of Non-Rectangular Prisms
In this lesson, you will investigate different strategies for finding the volume of non-rectangular
prisms. The problems require you to describe how you visualize each shape. As you work with
your team, ask each other these questions to focus your discussion:
How can the shape be broken down into simpler shapes?
Can we break the shape into equal layers or slices?
9-83.
THE SHELL BOX
Laurel keeps her seashell collection in a small box that is shaped
like a rectangular prism with a base of 4" by 3" and height of 5" (see
diagram at right). She wants to make a bigger box for her shells,
and she has found a pattern that she likes. The pattern has two sides
that are pentagons, and the finished box will look like a miniature
house (see picture at right). Use the net on the Lesson 9.2.4
Resource Page to construct a paper model of Laurel’s new box.
a.
Calculate the volume of Laurel’s original small box.
What method did you use? [ 3 4 5 = 60 cubic inches.
Find the area of the base and multiply by the height. ]
b.
Laurel was trying to figure out the volume of the new box.
5"
4"
3"
One of her friends said, “Why don’t you find the volume of one layer and then
figure out how many layers there would be?”
Another friend said, “What if you separated the pentagon into two parts, so that
you have a rectangular prism and a prism with a triangle base? Then you could
find the volume of the two parts separately and put them back together.”
Talk about each of these strategies with your team. Could Laurel use these ideas
to find the volume of her new box? [ Sample response: Yes, both strategies
work; to use the first strategy, you need to be careful about how you choose
one layer so that they are the same size through the whole shape. ]
Problem continues on next page.
Chapter 9: Circles and Volume
965
9-83.
Problem continued from previous page.
c.
9-84.
Work with your team to calculate the volume of
Laurel’s new box. Show your thinking clearly and be
prepared to share your method with the class.
[ New box volume calculated separately as
rectangular and triangular prisms =
(6 5) 8 + ( 624 ) 8 = 336 cubic inches, new box area
as product of pentagonal base and height =
(6 5 + 624 ) 8 = 336 cubic inches. ]
Find the volume of the prism at right using
any method you choose. Be prepared to
explain your reasoning. [ 3780 ft3 ]
5"
5"
9"
8"
6"
10 ft
12 ft
25 ft
38 ft
9-85.
9-86.
10 ft
Bryan was looking at the prism in problem 9-84 with his team. He wondered how
they could slice the prism horizontally into layers that would all be the same size.
a.
If the team slices the shape so that the bottom layer has the 10 m by 38 m
rectangle as its base, will each layer be equal in size? [ No ]
b.
How could the team turn or tip the shape so that if the shape is sitting on a table
and is cut horizontally, each layer will be the same size and shape? Which face
will rest on the table? [ The team could tip the shape so the front or back face
(trapezoid) rested on the table. ]
c.
When does it make sense to tip the shape in order to see the layers? Discuss this
question with your team and summarize your conclusion on your paper.
[ When the base is not the side resting on the table, tipping it can help
visualize the layers; some students may argue that it is never necessary to
tip it, as you can orient the layers vertically. ]
LEARNING LOG
The method for finding the volume of a prism is
commonly described as “finding the area of the base, then
multiplying by the height.” Discuss with your study team
how this strategy works. How does the area of the base
help to find the volume? Does it matter which face is the
base? Why do you multiply by the height? Record your
conclusions in your Learning Log. Title the entry,
“Volume of a Prism,” and label it with today’s date.
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Core Connections, Course 2