Lesson 4: Surface Area of 3-D Objects

Now that you are familiar with estimating and converting between SI (metric) units and imperial units, you will expand your knowledge to include 3-D objects that include curved surfaces. These objects include cylinders, spheres, cones, prisms, and pyramids. You will use the skills gained in Lesson 2 where you explored measuring curved surfaces.

Our surroundings are full of various 3-D objects, many of which can be broken into smaller, basic objects whose surface area and volume can be calculated. You will investigate surface area in this lesson, and by the end of this lesson you will be able to apply and use the surface area calculations needed in your project.

Outcomes

At the end of this lesson, you will be able to solve, using SI and imperial units, problems that involve the surface area of objects, including:

• right cones
• right cylinders
• prisms
• pyramids
• spheres

Lesson Questions

By the end of this lesson you should feel comfortable solving the following questions:

• How is the concept of surface area applied to understanding the design of structures?
• How do you determine the surface area of a 3-D object?

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 by using the Quizzes link under the Activities block.

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

Are you Ready?

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.

You will use this illustration to answer questions 1 and 2.

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

How did you do? Did you remember the difference between a prism and a pyramid? Did you remember how to find the area of basic shapes? Great work, if you remembered! If you did not remember, please carefully read this Refresher section.

Let’s take a look at what area means and how to find areas of basic shapes. This information will not only be helpful as you prepare to find the surface area of objects in this lesson but also when you explore the volume of those objects in the next lesson.

Area Definition

The Mathematics Glossary defines the term area. Go to “Area” to learn more. It contains an interactive Java applet and Flash animations to illustrate the definition. (Username is LA53, password is 4487 if needed.)

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Finding Area with Unit Squares

The mathematics lesson “Finding Area with Unit Squares” explores the measurement of square units. The lesson presents the formulas for the area of a rectangle, parallelogram, and triangle, and it includes math problems that involve the practical application of these formulas.(Username is LA53, password is 4487 if needed.)

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Estimating Area Using a Grid

The mathematics lesson “Estimating Area Using a Grid” uses inscribed polygons, circumscribed polygons, and the concept of limits to explain how the area of a circle can be measured. The lesson includes a game and a math problem that demonstrate the practical application of the formula for the area of a circle.

Materials

• paper
• prism net and pyramid net
• soup can with label
• scissors
• scotch tape
• a rectangular prism, such as a wooden block, a shoe box, or a cereal box

You will also require these materials to complete Math Lab: The Surface Area of an Orange:

• orange
• string
• ruler

Math Lab: The Surface Area of an Orange ** OPTIONAL TO COMPLETE**

Go to Math Lab: The Surface Area of an Orange, print it (or copy by hand) and complete it. You are asked to post to the discussion board L4: Discussion Board Posting - Orange Math Lab.

Keep this in your course folder (binder) as you will need to refer to it later in the lesson.

In order to complete the Math Lab, you will need to go to 1 cm × 1 cm Grid Paper and print a copy.

In this lesson the suggested glossary terms to add are the following:

• 3-D object
• apex
• lateral area
• net
• prism
• pyramid
• regular polygon
• right cone
• right cylinder
• sector
• sphere
• surface area

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Imagine taking a 3-D object and submerging it in a tub of water. The area of the object that is in contact with the water is called the surface area. You can also think of surface area as the measure of how much exposed area that a solid object has.

How would you determine the surface area of an object?

One way to do determine the surface area of the object is to peel off the outer layer of the object and then calculate the area of the peel. The peel is called the net. In Math Lab: The Surface Area of an Orange, you obtained a net of the orange (i.e., the orange peel sections) and added the areas of each part of the net (i.e., each peel section) to obtain the surface area of the orange.

As you move through this lesson, you will examine the nets of other 3-D objects. By adding together the sections of a net, you can determine the surface area of those objects.

Watch and Listen

Use "Surface Area of Prisms" to find out how to use the net of a rectangular prism to determine its surface area.

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then watch “Surface Area of Prisms 2 ” to see another example of the net of a rectangular prism used to determine its surface area. 

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Try This: Prisms, Pyramids, and Cylinders

Work with a partner (if possible) to examine the nets of prisms, pyramids, and cylinders. Use the following document titled Surface Area and Volume Investigation (or copy by hand) to summarize the information you will collect during this investigation. You may want to use the materials outlined in the Launch section as part of the investigation.

• paper
• prism net and pyramid net
• soup can with label
• scissors
• scotch tape
• a rectangular prism, such as a wooden block, a shoe box, or a cereal box
•  toblerone box if possible

For each 3-D object, create a net that can be easily folded into the 3-D object. To get started, you can use the following descriptions. However, you are not limited by the descriptions. You are free to use other ways of creating a net for the object.

Remember to add the information you discover to your “Surface Area and Volume Investigation” document. Then save a copy of your completed handout in your course folder.

Use the cereal box to create the net of a rectangular prism.

• Carefully unfold the cereal box and flatten it.
• Cut off any flaps that are not part of the surface area. (Some flaps will remain because they are a part of the exterior of the box.)

Use an empty Toblerone box to create the net of a triangular prism.

• Carefully unfold the box and flatten it.
• Cut off any flaps that are not part of the surface area.

To create a net of a square pyramid, go to Square Pyramid Net and print a copy.

• Cut out the net.
• Fold along the lines so it forms a three-dimensional pyramid.

Use the soup can with a label to create a net of a cylinder. For an idea of how you can do this, follow this procedure.

• Trace each circular end onto a sheet of paper.
• Cut out each of the ends.
• Remove the label from the soup can.
• Tape the circular ends to the label so that the net can be rolled into a 3-D cylinder.

Try This

TT 1. For each net, describe how many sections there are in the net.

TT 2. For each net, describe what dimensions will be needed to determine the areas of each section.

TT 3. For each net, describe what the formula for the surface area could look like. Try to match the formula below to each shape:

Use the link below to check your answers to Try This 1 - 3.

Possible TT1- 3 Solutions

Here is a great video that shows you how to find Surface Area of a Right Rectangular prism

Read

Read the textbook.

Self-Check

You have seen in the examples how a formula is applied in finding the surface area of a pyramid. Now use the formulas that you created to find the solutions to the following problems.

SC 1. Determine the surface area of the following pyramid, to the nearest in².

SC 2. Determine the surface area of the cylinder to the nearest cm².

Note the answer to SC2 is 150.8 cm². Typo in the answer.

SC 3. Determine the surface area of the following prism, rounded to the nearest cm².

SC 4. Consider the net shown. What does this net represent?

1. A cube with a side 10 cm.
2. A box with a length and width 10 cm and a height 5 cm.
3. A box with a length and width 5 cm and a height 10 cm
4. This net does not represent a box.

Compare your answers.

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

Project Connection **NOT ASSIGNED**

Think about the place that is the focus of your project. What 3-D objects are found in your place? Are there prisms, cones, cylinders, or spheres? If your place is fairly empty of objects, think of the 3-D objects that could occupy your place.

Now go to the Unit 1 Project and complete the Lesson 4 portion of your project.

Going Beyond

Did you know that taller trees generally have more leaves? The water in a tree needs to get from the roots to where photosynthesis happens, which is in the leaves and green parts. If you have a really tall tree, the plant has to force the water against gravity up to its leaves.

How is this possible? Consider the study of surface area. If a plant has large leaves, or numerous smaller leaves, then there is more surface area for evaporation to take place. In a plant, this is called transpiration. When transpiration occurs, the water leaving the plant is replaced by water coming up from the ground—and this water has dissolved nutrients in it.

For more information, initiate an Internet search using the keyword “transpiration” to see what else you can learn about a plant and the surface area of its leaves.

• How is the concept of surface area applied to understanding the design of structures?
• How do you determine the surface area of a 3-D object?

In this lesson you examined the nets of various 3-D objects. These objects included the prism, pyramid, cylinder, sphere, and cone. From the nets, you were able to determine the surface area formulas for each object. These formulas can be used to determine the surface area of any of the 3-D objects investigated, as long as the required information is given.

Whether you are figuring out how much paint you need to buy or how much wood is needed to build a shed, solving problems involving area comes in very handy.

In the next lesson you will investigate volume, another important concept in design. Then in Lesson 6 you will apply what you have learned about surface area and volume to solve problems.