Lesson 1

Lesson 1: Area and Surface Area

 

Focus

 

The bark canoe was first used by First Nations people across Canada. This canoe was well suited for travel on lakes or rivers. The crafts were later used by voyageurs to transport furs and other goods between trading posts during European exploration of North America. The basic design was lightweight and manoeuvrable, yet the canoe was sturdy enough to carry heavy loads. The canoe's basic design has stood the test of time.

 

There are still a few dedicated canoe artisans who follow traditional construction techniques. In the photograph, the birchbark cover is carefully stitched together. The amount of bark needed depends on the canoe’s surface area.

 

Henri Vaillancourt

 

In this lesson you will explore the surface area of a variety of three-dimensional objects.

 

Lesson Question

• How can a net be used to explain the relationship between area and surface area?

Assessment

 

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3 Glossary Terms and the Formula Sheet
  
  
• Lesson 1 Assignment (Save a copy of your lesson assignment document to your course folder now.)
  
  
• the Project Connection

In this course you may come across Self-Check questions, Try This questions, and other activities that may or may not be assessed.

 

Remember that these questions and activities provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance.

 

Materials and Equipment

• paper
• scissors
• tape or a glue stick
• calculator

Time

 

This lesson has been designed to take 150 minutes; however, it may take more or less time depending on how well you are able to understand the lesson concepts. It is important that you progress at your own pace based on your own learning needs.

 

Launch

 

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 1.

 

Complete the following Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

 

Are You Ready?

 

In previous mathematics courses, you found the area of various two-dimensional shapes. Hopefully you recall using these formulas.

 

1. Find the area of one side of a sheet of standard copy paper. Express your answer to the nearest tenth of a square inch. Answer
   
   
   iStockphoto/Thinkstock
   
   
2. A circular no-entry sign is 45 cm in diameter. What is the area of the sign’s front side? Round your answer to the nearest square centimetre. Answer
   
   
   iStockphoto/Thinkstock
   
   
3. The trapezoidal window outlined in blue on the photo has the dimensions outlined by the graphic. What area of glass is needed to fill the window? Answer
   
   
   iStockphoto/Thinkstock
   

If you answered the Are You Ready? questions without any issues, move on to Discover.

 

If you had some difficulty with the Are You Ready? questions, complete Refresher.

 

Refresher

 

If you had trouble with the Are You Ready? questions, or if you require more information, use the following activites to guide your review of area formulas.

 

	Area Formulas Mathematics 10-3: Unit 2: Module 4 Lessons 2 and 3 cover the use of the various area formulas.
	Use the “Circles — Area Perspectives” link, then push the "Topics" button, and explore the sections "Area of a Circle Formula" and "Applying the Formula."

 

Go back to Are You Ready?, and try the questions again. Contact your teacher if you continue to have difficulty with the questions.

 

Discover

 

In Module 5, one important question asked when making models was this: How much material is needed? If you are referring to the object’s surface, this question is really asking this: What is the surface area of the model?

 

In Exploring Surface Area, Volume, and Nets—Use It, you will investigate what surface area means. Click on the button and use the information to complete Try This 1.

 

Try This 1

 

Select the “Use It 1: Nets” tab of the applet to begin each investigation.

 

Investigate the following three-dimensional objects:

• rectangular prism
• rectangular pyramid
• triangular pyramid

Answer the following questions for each object. You can organize your responses in a chart similar to the one that follows.

1. Select a 3-D object from the area showing all six objects, and then choose the correct net for that object from the choices given. (Hint: Use the slider from the applet to help identify the correct net.) Sketch the net in your chart.
   
   
2. Use either the “Use It 1: Nets” or “Use It 2: Views” tab from the applet to help answer these questions:
   1. How many faces does the net have? 
   2. What 2-D shape does each face have?
      
      
3. What 2-D area formula could you use to find the area of each face?
   
   
4. Describe a strategy you could use to find the total surface area of the 3-D object.
   
   

   	Rectangular Prism	Rectangular Pyramid	Triangular Pyramid
   Sketch of Net
   Number and Shape of Each Face
   Formulas Used to Find the Area of Each Face
   Strategy Used to Find the Total Surface Area of the 3-D Object

Save your responses to your course folder.

 

Share 1

 

Share your responses to the Try This 1 questions with a partner or with a group of people. How did the strategies identified to find the surface area of the 3-D object compare? After comparing responses, discuss the following questions:

• What is the difference between an area of a 2-D shape and the surface area of a 3-D object?
  
  
• How can a net be used to help find the surface area of a 3-D object?

As you work on Share 1, refer to the Share Rubric. This information will help you understand what your teacher is expecting.

 

If required, save a copy of your discussion in your course folder.

 

Explore

 

In previous mathematics courses, much of your work with area has involved two-dimensional shapes. In this lesson your focus will be on the surface area of three-dimensional objects. As you may have noted in Discover, the surface area of a 3-D solid is the sum of the areas of all the solid's sides, faces, or exposed surfaces.

 

In Lesson 1 you will explore the surface area of a variety of different 3-D solids. You will begin by exploring the surface area of cylinders, such as the round bales shown in the photo.

 

Try This 2

 

To prevent spoilage in feed for animals, one option for farmers is to create silage bales. A silage bale is a cylindrical bale of hay or grain covered in plastic. This covering prevents weathering and creates silage, partially fermented feed. Silage is rich in nutrients and can be used as feed during winter. Bales range in size, but you can assume the bales in the photograph are 5 ft in diameter × 5 ft long.

 

To know how much plastic to buy for the silage, the farmer must first know the surface area of each bale.

 

One way to determine the surface area of each round bale is to draw a net.

 

1. Fill in the values for width and diameter on the net diagram.
   
   
2. How is the length of the cylindrical side related to each of the bale’s circular ends? Calculate this length, and fill in the information on the net.
   
   
   
   

3. Calculate, in square feet, the area of the side (the rectangle) of the bale. Round to one decimal place.
   
   

4. Find the area of each circular end of the bale. Round your answer to one decimal place.
   
   
5. Determine, to one decimal place, the bale’s surface area.
   
   
6. Use the surface area of a cylinder formula to check your answer to question 5. How do your answers compare?
   
   
7. If the farmer has 100 bales, how much plastic will need to be purchased? Round your answer to the nearest square foot.

Save your work in your course folder.

 

Self-Check 1

 

The following example explores the surface area of a triangular prism. Before you read through the example, use the Exploring Surface Area, Volume, and Nets applet from Try This 3 to view the net and the surface area formula for a triangular prism.

 

Example

 

Anton and Xera have volunteered to paint over the graffiti on a skateboard ramp. They took measurements of the ramp and noticed that the top, back, and two sides could be approximated by a triangular prism.

 

Xera and Anton sketched the prism and used the information to calculate the surface area they need to paint. They plan to give the ramp two coats. What is the total area that they must buy paint for? Note: The bottom will not be painted.

 

iStockphoto/Thinkstock

 

Solution

 

Draw the net with dimensions showing the four sides that need to be painted.

 

SA = (area of rectangular back) + (area of square top) + (2 × area of triangular side)

 

 

 

Since Anton and Xera want to apply two coats of paint, they need to buy enough material to cover an area of 2 × 52 ft2 = 104 ft2.

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Self-Check 2

1. A wooden window box is constructed from three rectangular pieces of wood and two identical trapezoidal end pieces. What surface area needs to be painted if you only paint the outside of the box?
   
   
   iStockphoto/Thinkstock
   
   Draw a net of the planter. Then, determine the surface area of the outside of the planter. Answer

Connect

 

Going Beyond

 

The surface area of a cone can also be determined by finding the area for each shape in the cone's net. The area formulas for the shapes in the net are used to develop the surface area formula for the cone. The formula for the surface area of a cone is developed below.

 

You may find it helpful to have a 3-D model of a cone as you work through the following steps. You can print the net of a cone to help build your 3-D model.

 

Call the radius of the cone’s circular base r and the slant height of the side h. The slant height is measured along the cone’s surface from the base to the top of the cone.

 

Now compare your 3-D cone to its net.

 

 

Look at the net. The red portion of the large circle is the surface of the cone.

 

The large circle of radius s has a circumference of and an area of .

 

But part of that circle is missing!

 

The part of the large circle coloured red is formed from an arc that is exactly as long as the circumference of the small circle. The circumference of the small circle is .

 

Now look at the circumference of the large circle. Compare the red part of the circumference to the whole circumference. You find the following fraction.

 

 

You can use this ratio to calculate the area of the red portion of the cone to be

 

  So, the surface area of the cone = area of the conical side + area of the circular base.

 

 

Project Connection

 

The Module 6 Project is actually a combined project for Modules 6 and 7. Take some time to familiarize yourself with the project goals and requirements. You will begin working on the project steps in Lesson 4 or when you feel comfortable with surface area.

 

Lesson 1 Assignment

 

Lesson 1 Summary

 

 

Umiaks were large open boats used by the Inuit in the Canadian Arctic. The frames were fashioned from driftwood and were covered with seal or walrus skins. It is likely the Inuit a thousand years ago used boats similar to the one in the photograph in their migration across the Arctic Ocean from Eastern Siberia. 

 

The quantity of skins required to cover the umiak’s frame depended on the surface area of the frame. In this lesson you explored surface area and its relationship to the nets of three-dimensional objects.