Lesson 4: Surface Area of 3-D Objects

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Review the results you obtained for your Math Lab data and analysis. In your analysis of the results, you evaluated the ratio . You also looked at what other students obtained for this ratio. Did you find that the ratio was close to 3.14? In fact, under ideal circumstances, this ratio would be π.

TT 4. What might be some reasons why someone doing the Math Lab did not get a number that was close to π?

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Now, see what happens if you take the ratio and rearrange it.

You can take the formula further by substituting d = 2r.

This is the formula for the surface area of a sphere, in terms of the radius.

Add this formula to your list of formulas. You should save your list of formulas to your course folder.

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Foundations and Pre-calculus Mathematics 10 (Pearson)
Read “Example 1: Determining the Surface Area of a Sphere” on page 47 to see how the formula A = 4r² is used to calculate the surface area of a sphere. In the question shown in this example, the radius of the sphere is not given. Read the solution and think of another formula that could be used to solve the problem.Read “Example 2: Determining the Diameter of a Sphere” on pages 47 and 48 to understand how to use the surface area of a lacrosse ball to determine the diameter of the ball. Pay attention to how the formula is rearranged to find the desired result.

Self-Check

SC 5. Find the surface area of the following sphere to the nearest square metre.

SC 6. Determine the radius of a sphere with a surface area of 64cm². Report your answer to the nearest centimetre.

Compare your answers.

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From the examples, you have learned that you can determine the surface area of a sphere using the formula A = 4r². You have also seen how the formula can be used to determine the radius of a sphere, if you know the surface area.

Practice what you have learned by completing TT 5 in your course folder ( binder).

Foundations and Pre-calculus Mathematics 10 (Pearson) textbook

TT 5. Complete “Exercises” questions 3.a), 3.c), 8, 9, 13.a), 13.b), and 15 on pages 51 and 52.

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There are other ways of determining the surface area of 3-D objects besides analyzing their nets. Often in mathematics, you can discover properties of unfamiliar objects by examining the properties of familiar ones.

For example, the cone is a 3-D object that is shaped much like a pyramid. Like a pyramid, a cone has only one base and the lateral faces of the cone meet at a point called the apex.

The illustration above shows that as you increase the number of sides on the base, the number of faces also increases. The area of each face also becomes smaller.

Eventually, the polygon base approaches the shape of a circle and the lateral area of the pyramid approaches the lateral area of the cone.

You can figure out the formula for the surface area of a cone with this idea in mind.

Consider the formula for the surface area of a rectangular pyramid, as shown in the illustration. The height of the triangular faces, or slant height, is labelled s. The sides of the base are labelled a, b, c, and d.

In the case of a cone, the perimeter of the base is really the circumference of a circle, so its surface area formula would be

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Foundations and Pre-calculus Mathematics 10 (Pearson)

Read “Example 3: Determining the Surface Area of a Right Cone” on page 32 to see how the formula is used to calculate the surface area of a right cone. Pay careful attention to how the slant height of the cone is determined. What theorem is used?

Self-Check

Now that you have watched some videos and had a chance to talk with your classmates, it is your turn to try some Self-Check questions to see if you have figured out surface area.

SC 7. When assembled, the net in the preceding illustration will create a

1. cube
2. cylinder
3. cone
4. prism

SC 8. Determine the surface area of the following cone to the nearest square foot.

Compare your answers.

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Complete the following questions in your course folder ( binder).

Foundations and Pre-calculus Mathematics 10 (Pearson)

TT 6. Complete “Exercises” questions 7, 12, and 16.a) on pages 34 and 35.

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