Lesson 5: The Volume of 3-D Objects

Have you ever moved from one residence to another? A move can take a great deal of planning, co-ordination, and time.

You may need to rent a large truck to contain all of your belongings, or you may have to temporarily store your belongings in a storage facility like the one pictured. The more furniture, clothing, and other possessions that you need to transport, the larger the truck or storage unit that you need for the move.

The amount of space becomes an important consideration. This is known as volume. In this lesson you will investigate the concept of volume and learn how to determine the volume of 3-D shapes.

Outcomes

At the end of this lesson, you will be able to determine the volume of a right cone, a right cylinder, a right prism, a right pyramid, or a sphere using an object or its labelled diagram.

Lesson Questions

• How is the concept of volume applied to understanding the design of structures?

• How are the formulas for the volumes of solids related to each other?

Lesson Completion and Assessment

As you work through each lesson, complete all the questions and learning activities in your binder using paper and pencil, clearly labeling your work (they refer to this as your course folder). These include the Are you Ready, Try This, Share and Self Check questions. Check your work if answers are provided. Remember that these questions provide you with the practice and feedback that you need to successfully complete this course.
Once you have completed all of the learning activities, take the Lesson Quiz. This is the assessment for each lesson and is located under the Assess tab or using the Quizzes link under the Activities block.

** Note – Share questions may have to be done on your own depending on your learning situation**

Complete these questions in your course folder (binder). If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

1. What are some differences between area and volume?

2.   What is the formula for the volume of a rectangular prism?

3.   Find the volume of a cube with side length of 5 cm.

4.   Find the volume of a rectangular prism with a length of 14 m, a width of 3 m, and a height of 2 m.

5.   How could you find the volume of a 3-D object without using a formula?

Once you have completed these exercises to the best of your ability, use the provided answer link to check your work.

Answers

If you feel comfortable with the concepts covered in the questions, move forward to Discover. If you experienced difficulties or want more practice, use the resources in Refresher to review these important concepts before continuing through the lesson or contact your teacher.

Refresher

Volume

This resource from the Mathematics Glossary defines the term volume. Go to “Volume" by clicking below.
You will find an animation to illustrate the definition. (Usename is LA53, password is 4487 if needed)

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Volume and Displacement

In the interactive mathematics lesson titled “Volume and Displacement,” you can calculate the volume of rectangular prisms. You will also learn that the volume of an irregular object can be found by measuring the amount of water the object displaces. (Usename is LA53, password is 4487 if needed)

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Go to Math Lab: Comparing the Volume of a Cylinder and a Sphere, print it (or copy by hand) and complete it.

Keep this in your course folder (binder) to refer to later.

© skvoor/shutterstock

Glossary Terms

Add the following terms to your “Glossary Terms” section to your notes:

• area

• base

• volume

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Recall that area is the amount of square units occupying an enclosed shape or two-dimensional space. To find the area of a shape, you need to multiply two dimensions of the shape together. For example,

1 cm × 1 cm = 1 cm²

On the other hand, volume measures the amount of cubic units occupying a three-dimensional space. To find the volume of an object, you need to multiply three dimensions of the object together. For example,

1 cm × 1 cm × 1 cm = 1 cm³

Prisms

Watch and Listen

Volume of a Prism

Watch the multimedia presentation titled “Volume of a Prism.” See if you can remember the three steps that are needed to find the volume of a prism.

Generally speaking, the formula for the volume of a prism is the following:

volume = area of base × height

Self-Check

Use the formula for the volume of prisms to solve the following problems.

SC 1. Find the volume of the triangular prism shown here.

SC 2. Find the volume of a rectangular prism that has a base measuring 6 in by 4 in and a height of 8 in.

Compare your answers.

Cylinders

A cylinder is a prism with a circular base. Although you might not call a cylinder a circular prism, that’s exactly what a cylinder is.

Using the formula for the volume of a prism, what could be the formula for the volume of a cylinder?

Save your answer to your course folder (binder); then check in your textbook to see if your answer is correct. If it isn’t correct, what parts of your formula were correct? What parts of the formula need to be changed?

Here are a few examples to look over.

Example 1

Determine the volume of a soup can with a diameter of 3.5 in and a height of 4.5 in. Show your solution to the nearest tenth of a square inch.

Solution

Example 2

The volume of a cylinder is 200 cm3. Determine the radius of the cylinder to the nearest hundredth of a centimetre if the height of the cylinder is 8 cm.

Solution

Try This

Practice using the formula for the volume of a cylinder to solve these problems by completing them in your course folder ( binder).

Foundations and Pre-calculus Mathematics 10 (Pearson)

TT 1. Complete “Exercises” questions 6 and 18.c) on pages 42 and 43.

Use the link below to check your answers to Try This 1.

Possible TT1 Solutions

Cones

Math Lab: Volume of a Cone

Go to Math Lab: Volume of a Cone, print it (or copy by hand) and complete it.

Keep this in your course folder (binder) to refer to later.

Read

Go to your textbook to find the formulas for a cylinder and a cone and note how they are similar and how they are different. What fraction of the volume of a cylinder is the volume of a cone with the same height and radius?

Compare your experimental result in the cone investigation with the formulas you have just read in your textbook. Is your experimental result confirmed? Do your results support the finding that the volume of a cone is the volume of a cylinder with the same height and radius?

If your results do not support the ratio, give some reasons why you think this might be the case. Incorporate these comments into your Math Lab.

Read

KEY IDEA

You have learned that the volume of a cone is the volume of a cylinder with the same radius and height. This ratio is the same for pyramids. In other words, the volume of a pyramid is the volume of the prism with the same height and base area.

Read

Take a look at the following example, which compares the relationship between the volume of a right rectangular pyramid and the volume of a right rectangular prism.

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Self-Check

SC 3. Find the volume of the square pyramid shown in the diagram.

Compare your answers.

Spheres

Retrieve your analysis from Math Lab: Comparing the Volume of a Cylinder and a Sphere that you saved to your course folder.

The height of the can is equal to twice the radius of the ball.

To determine the formula for the volume of a sphere, you can do what you did for the cone. First, what was the ratio of the volume of the tennis ball compared to the volume of the juice container? Did you observe that the volume of the tennis ball was about that of the container?

This means that the volume of the sphere would be .

This formula is correct, but there’s a way to simplify the formula by finding another way to express the can’s height.

Think about the fact that the can and the ball have the same radius and the same height. Does it make sense to you that the height of the can would be equal to twice the radius of the ball?

So you could write the volume of a sphere as or, more simply, .

Read

Try This

Complete TT 2 in your course folder ( binder) to practise applying the formula for the volume of the sphere. You may have to do something more than apply the formula for the context-based questions.

Foundations and Pre-calculus Mathematics 10 (Pearson)

TT 2. Complete “Exercises” questions 13.c), 13.d), and 19 on pages 51 and 52.

Use the link below to check your answers to Try This 2.

Possible TT2 Solutions

The following table summarizes the different types of 3-D objects you have examined in this lesson. Included are the formulas that have been developed for these objects.

Self-Check

SC 4. Sheila is excavating the basement for her house on a small lot in town. The dimensions of the basement excavation need to be 40-ft long by 30-ft wide and 9-ft deep. The excavated soil is placed in a circular area beside the excavation. The radius of this area is 20 ft. As more soil is added, the soil pile forms the shape of a cone. The highest the excavator can lift the soil is 24 ft.

Backhoe © ownway/shutterstock

Will the excavator be able to put all the soil from this excavation into this one cone-shaped pile? (Show your calculations to the nearest whole number.)

Compare your answers.

Lesson Assessment

Complete the lesson quiz posted under the Assess tab or using the Quizzes link under the Activities block. Also ensure your work in your binder (course folder) is complete. 

Project Connection **NOT ASSIGNED**

In this lesson you are ready to determine the volume of some of the shapes that make up your place.

You should go to the Unit 1 Project and complete the Lesson 5 portion of the project.

Going Beyond

Thanks to advances in technology, the world is truly changing. You can have a different perspective on the concept of place by using Google Earth.

Initiate an Internet search using the keywords “real world math” and “volume of solids.” Using these search terms, you should find a website titled Real World Math. There’s an exercise titled “Volume of Solids” that shows you how to determine the surface area and volume of some of the world’s more famous places.

In this lesson you examined the following questions:

• How is the concept of volume applied to understanding the design of structures?

• How are the formulas for the volumes of solids related to each other?

In this lesson you looked in your surroundings for various three-dimensional shapes including right cones, right cylinders, prisms, pyramids, and spheres. You discovered the relationships between the volumes of related 3-D objects through hands-on labs and by using interactive activities.

You developed strategies for determining the volume of a right cone, a right cylinder, a right prism, a right pyramid, or a sphere using an object or its labelled diagram.

In the next lesson you will use what you have learned about surface area and volume to solve problems in real-world situations.