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Volume of Composite Three-Dimensional Objects

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Examples, videos, and solutions to help Grade 7 students learn how to compute volumes of three-dimensional objects composed of right prisms by using the fact that volume is additive.

New York State Common Core Math Grade 7, Module 6, Lesson 26

Worksheets for 7th Grade, Module 6, Lesson 26 (pdf)

Lessons 26 Student Outcomes

• Students compute volumes of three-dimensional objects composed of right prisms by using the fact that volume is additive.

Lessons 26 Summary

• To find the volume of a three-dimensional composite object, two or more distinct volumes must be added together (if they are joined together) or subtracted from each other (if one is a missing section of the other).

• There are two strategies to find the volume of a prism: Find the area of the base, then multiply times the prism’s height; decompose the prism into two or more smaller prisms of the same height and add the volumes of those smaller prisms.

Lesson 26 Classwork

Example 1
Find the volume of the following three-dimensional object composed of two right rectangular prisms.

Exercise 1
Find the volume of the following three-dimensional figure composed of two right rectangular prisms.

Exercise 2
The right trapezoidal prism is composed of a right rectangular prism joined with a right triangular prism. Find the volume of the right trapezoidal prism shown in the diagram using two different strategies.

Example 2
Find the volume of the right prism shown in the diagram whose base is the region between two right triangles. Use two different strategies.



Example 3
A box with a length of 2 ft., a width of 1.5 ft., and a height of 1.25 ft. contains fragile electronic equipment that is packed inside a larger box with three inches of styrofoam cushioning material on each side (above, below, left side, right side, front, and back).
a. Give the dimensions of the larger box.
b. Design styrofoam right rectangular prisms that could be placed around the box to provide the cushioning; i.e., give the dimensions and how many of each size are needed.
c. Find the volume of the styrofoam cushioning material by adding the volumes of the right rectangular prisms in the previous question.
d. Find the volume of the styrofoam cushioning material by computing the difference between the volume of the larger box and the volume of the smaller box.

Example 1

Find the volume of the following three-dimensional object composed of two right rectangular prisms.

Example 2

Calculate the volume of the following prism.

Example 3

A container is shaped like a right pentagonal prism with an open top. When a cubic foot of water is dumped into the container, the depth of the water is inches. Find the area of the pentagonal base.

Example 4

Find the volume of the right prism shown in the diagram whose base is the region between two right triangles.

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