# Volume and Surface Area

Videos and solutions to help Grade 7 students learn how to solve real-world and mathematical problems involving volume and surface areas of three- dimensional objects composed of cubes and right prisms.

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## New York State Common Core Math Grade 7, Module 3, Lesson 26

### Lesson 26 Student Outcomes

• Students solve real-world and mathematical problems involving volume and surface areas of three- dimensional objects composed of cubes and right prisms.

Lesson 26 Classwork

Opening Exercise
Explain to your partner how you would calculate the area of the shaded region. Then, calculate the area.

Example 1: Volume of a Shell
The insulated box shown is made from a large cube with a hollow inside that is a right rectangular prism with a square base. The figure at right is what the box looks like from above.
a. Calculate the volume of the outer box.
b. Calculate the volume of the inner prism.
c. Describe in words how you would find the volume of the insulation.
d. Calculate the volume of the insulation in cubic centimeters.
e. Calculate the amount of water the box can hold in liters.

Exercise 1: Designing a Brick Planter
You have been asked by your school to design a brick planter that will be used by classes to plant flowers. The planter will be built in the shape of a right rectangular prism with no bottom so water and roots can access the ground beneath. The exterior dimensions are to be 12 ft. x 9 ft. x 2 1/2 ft. The bricks used to construct the planter have are 6 inches long, 3 1/2 inches wide, and 2 inches high.
a. What are the interior dimensions of the planter if the thickness of the planter’s walls is equal to the length of the bricks?
b. What is the volume all the brick that forms the planter?
c. If you are going to fill the planter 3/4 full of soil, how much soil will you need to purchase, and what will be the height of the soil?
d. How many bricks are needed to construct the planter?
e. The bricks used in this project cost \$0.82 each and weigh 4.5 lb. each. The supply company charges a delivery fee of \$15 per whole ton (2000 lb.) over 4000 pounds. How much will your school pay for the bricks (including delivery) to construct the planter?
f. A cubic foot of top soil weighs between 75 and 100 lbs. How much will the soil in the planter weigh?
g. If the topsoil costs \$.88 per each cubic foot, calculate the total cost of materials that will be used to construct the planter.

Exercise 2: Design a Feeder
You did such a good job designing the planter that a local farmer has asked you to design a feeder for the animals on his farm. Your feeder must be able to contain at least 10,000 cubic centimeters, but not more than 20,000 cubic centimeters of grain when it is full. The feeder is to be built of stainless steel and must be in the shape of a right prism, but not a right rectangular prism. Sketch your design below including dimensions. Calculate the volume of grain that it can hold and the amount of metal needed to construct the feeder.
The farmer needs a cost estimate. Calculate the cost of constructing the feeder if 1/2 cm thick stainless steel sells for \$93.25 per square meter.

Closing
• Describe the process of finding the volume of a prism shell.
• Find the volume of the outer figure; then, subtract the volume of the inner figure.
• How does the thickness of the shell affect the internal dimensions of the prism? The internal volume? The external volume?
• The thicker the shell, the smaller the internal dimensions and the smaller the internal volume. The external volume is not affected by the thickness of the shell.

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