Lesson 17: the Area of a Circle

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 7•3

Lesson 17: The Area of a Circle

Student Outcomes . Students give an informal derivation of the relationship between the circumference and area of a circle. . Students know the formula for the area of a circle and use it to solve problems.

Lesson Notes . Remind students of the definitions for circle and circumference from the previous lesson. The Opening Exercise is a lead-in to the derivation of the formula for the area of a circle. . Not only do students need to know and be able to apply the formula for the area of a circle, it is critical for them to also be able to draw the diagram associated with each problem in order to solve it successfully. . Students must be able to translate words into mathematical expressions and equations and be able to determine which parts of the problem are known and which are unknown or missing.

Classwork Exercises 1–3 (4 minutes)

Exercises 1–3

Solve the problem below individually. Explain your solution.

1. Find the radius of a circle if its circumference is ퟑퟕ. ퟔퟖ inches. Use 흅 ≈ ퟑ. ퟏퟒ.

If 푪 = ퟐ흅풓, then ퟑퟕ. ퟔퟖ = ퟐ흅풓. Solving the equation for 풓,

ퟑퟕ. ퟔퟖ = ퟐ흅풓 ퟏ ퟏ ( ) ퟑퟕ. ퟔퟖ = ( ) ퟐ흅풓 ퟐ흅 ퟐ흅 ퟏ (ퟑퟕ. ퟔퟖ) ≈ 풓 ퟔ. ퟐퟖ ퟔ ≈ 풓

The radius of the circle is approximately ퟔ 퐢퐧.

2. Determine the area of the rectangle below. Name two ways that can be used to find the area of the rectangle.

The area of the rectangle is ퟐퟒ 퐜퐦ퟐ. The area can be found by counting the square units inside the rectangle or by multiplying the length (ퟔ 퐜퐦) by the width (ퟒ 퐜퐦).

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 7•3

3. Find the length of a rectangle if the area is ퟐퟕ 퐜퐦ퟐ and the width is ퟑ 퐜퐦.

If the area of the rectangle is 퐀퐫퐞퐚 = 퐥퐞퐧퐠퐭퐡 ∙ 퐰퐢퐝퐭퐡, then

ퟐퟕ 퐜퐦ퟐ = 퐥 ∙ ퟑ 퐜퐦 ퟏ ퟏ ⋅ ퟐퟕ 퐜퐦ퟐ = ⋅ 풍 ⋅ ퟑ 퐜퐦 ퟑ ퟑ ퟗ 퐜퐦 = 풍

Exploratory Challenge (10 minutes)

Complete the exercise below. Scaffolding: Provide a circle divided into 16 Exploratory Challenge equal sections for students to To find the formula for the area of a circle, cut a circle into ퟏퟔ equal pieces. cut out and reassemble as a rectangle.

Arrange the triangular wedges by alternating the “triangle” directions and sliding them together to make a MP.7 “parallelogram.” Cut the triangle on the left side in half on the given line, and slide the outside half of the triangle to the other end of the parallelogram in order to create an approximate “rectangle.”

Move half to the other end.

The circumference is ퟐ흅풓, where the radius is 풓. Therefore, half of the circumference is 흅풓.

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What is the area of the “rectangle” using the side lengths above?

The area of the “rectangle” is base times height, and, in this case, 푨 = 흅풓 ∙ 풓.

Are the areas of the “rectangle” and the circle the same?

Yes, since we just rearranged pieces of the circle to make the “rectangle,” the area of the “rectangle” and the area of the circle are approximately equal. Note that the more sections we cut the circle into, the closer the approximation.

If the area of the rectangular shape and the circle are the same, what is the area of the circle?

The area of a circle is written as 푨 = 흅풓 ∙ 풓, or 푨 = 흅풓ퟐ.

Example 1 (4 minutes)

Example 1

Use the shaded square centimeter units to approximate the area of the circle.

What is the radius of the circle?

ퟏퟎ 퐜퐦

What would be a quicker method for determining the area of the circle other than counting all of the squares in the entire circle? ퟏ Count of the squares needed; then, multiply that by four in order to determine the area of the entire circle. ퟒ

Using the diagram, how many squares were used to cover one-fourth of the circle?

The area of one-fourth of the circle is approximately ퟕퟗ 퐜퐦ퟐ.

What is the area of the entire circle?

푨 ≈ ퟒ ∙ ퟕퟗ 퐜퐦ퟐ 푨 ≈ ퟑퟏퟔ 퐜퐦ퟐ

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Example 2 (4 minutes)

Example 2

A sprinkler rotates in a circular pattern and sprays water over a distance of ퟏퟐ feet. What is the area of the circular region covered by the sprinkler? Express your answer to the nearest square foot.

Draw a diagram to assist you in solving the problem. What does the distance of ퟏퟐ feet represent in this problem?

The radius is ퟏퟐ feet.

What information is needed to solve the problem?

The formula to find the area of a circle is 푨 = 흅풓ퟐ. If the radius is ퟏퟐ 퐟퐭., then 푨 = 흅 ∙ (ퟏퟐ 퐟퐭. )ퟐ = ퟏퟒퟒ흅 퐟퐭ퟐ, or approximately ퟒퟓퟐ 퐟퐭ퟐ.

Make a point of telling students that an answer in exact form is in terms of 휋, not substituting an approximation of pi.

Example 3 (4 minutes)

Example 3

Suzanne is making a circular table out of a square piece of wood. The radius of the circle that she is cutting is ퟑ feet. How much waste will she have for this project? Express your answer to the nearest square foot.

Draw a diagram to assist you in solving the problem. What does the distance of ퟑ feet represent in this problem?

The radius of the circle is ퟑ feet.

What information is needed to solve the problem?

The area of the circle and the area of the square are needed so that we can subtract the area of the circle from the area of the square to determine the amount of waste.

What information do we need to determine the area of the square and the circle?

Circle: just radius because 푨 = 흅풓ퟐ Square: one side length

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How will we determine the waste?

The waste is the area left over from the square after cutting out the circular region. The area of the circle is 푨 = 흅 ∙ (ퟑ 퐟퐭. )ퟐ = ퟗ흅 퐟퐭ퟐ ≈ ퟐퟖ. ퟐퟔ 퐟퐭ퟐ. The area of the square is found by first finding the diameter of the circle, which is the same as the side of the square. The diameter is 풅 = ퟐ풓; so, 풅 = ퟐ ∙ ퟑ 퐟퐭. or ퟔ 퐟퐭. The area of a square is found by multiplying the length and width, so 푨 = ퟔ 퐟퐭. ∙ ퟔ 퐟퐭. = ퟑퟔ 퐟퐭ퟐ. The solution is the difference between the area of the square and the area of the circle, so ퟑퟔ 퐟퐭ퟐ − ퟐퟖ. ퟐퟔ 퐟퐭ퟐ ≈ ퟕ. ퟕퟒ 퐟퐭ퟐ.

Does your solution answer the problem as stated?

Yes, the amount of waste is ퟕ. ퟕퟒ 퐟퐭ퟐ.

Exercises 4–6 (11 minutes) Solve in cooperative groups of two or three.

Exercises 4–6

4. A circle has a radius of ퟐ cm. a. Find the exact area of the circular region.

퐀 = 훑 ∙ (ퟐ 퐜퐦)ퟐ = ퟒ훑 퐜퐦ퟐ

b. Find the approximate area using ퟑ. ퟏퟒ to approximate 훑.

퐀 = ퟒ 퐜퐦ퟐ ∙ 훑 ≈ ퟒ 퐜퐦ퟐ ∙ ퟑ. ퟏퟒ ≈ ퟏퟐ. ퟓퟔ 퐜퐦ퟐ

5. A circle has a radius of ퟕ cm. a. Find the exact area of the circular region.

퐀 = 훑 ∙ (ퟕ 퐜퐦)ퟐ = ퟒퟗ훑 퐜퐦ퟐ

ퟐퟐ b. Find the approximate area using to approximate 훑. ퟕ ퟐퟐ 퐀 = ퟒퟗ ∙ 훑 퐜퐦ퟐ ≈ (ퟒퟗ ∙ ) 퐜퐦ퟐ ≈ ퟏퟓퟒ 퐜퐦ퟐ ퟕ

c. What is the circumference of the circle?

퐂 = ퟐ훑 ∙ ퟕ 퐜퐦 = ퟏퟒ훑 퐜퐦 ≈ ퟒퟑ. ퟗퟔ 퐜퐦

6. Joan determined that the area of the circle below is ퟒퟎퟎ흅 퐜퐦ퟐ. Melinda says that Joan’s solution is incorrect; she believes that the area is ퟏퟎퟎ흅 퐜퐦ퟐ. Who is correct and why?

Melinda is correct. Joan found the area by multiplying 흅 by the square of ퟐퟎ 퐜퐦 (which is the diameter) to get a result of ퟒퟎퟎ흅 퐜퐦ퟐ, which is incorrect. Melinda found that the radius was ퟏퟎ 퐜퐦 (half of the diameter). Melinda multiplied 흅 by the square of the radius to get a result of ퟏퟎퟎ흅 퐜퐦ퟐ.

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Closing (4 minutes) . Strategies for problem solving include drawing a diagram to represent the problem and identifying the given information and needed information to solve the problem. . Using the original circle in this lesson, cut it into 64 equal slices. Reassemble the figure. What do you notice?  It looks more like a rectangle. Ask students to imagine repeating the slicing into even thinner slices (infinitely thin). Then, ask the next two questions. . What does the length of the rectangle become?  An approximation of half of the circumference of the circle . What does the width of the rectangle become?  An approximation of the radius 1 . Thus, we conclude that the area of the circle is 퐴 = 퐶푟. 2 1 1  If 퐴 = 퐶푟, then 퐴 = ∙ 2휋푟 ∙ 푟 or 퐴 = 휋푟2. 2 2  Also see video link: http://www.youtube.com/watch?v=YokKp3pwVFc

Relevant Vocabulary

CIRCULAR REGION (OR DISK): Given a point 푪 in the plane and a number 풓 > ퟎ, the circular region (or disk) with center 푪 and radius 풓 is the set of all points in the plane whose distance from the point 푪 is less than or equal to 풓.

The boundary of a disk is a circle. The area of a circle refers to the area of the disk defined by the circle.

Exit Ticket (4 minutes)

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Name ______Date______

Lesson 17: The Area of a Circle

Exit Ticket

Complete each statement using the words or algebraic expressions listed in the word bank below.

1. The length of the of the rectangular region approximates the length of the of the circle.

2. The of the rectangle approximates the length of one-half of the circumference of the circle.

3. The circumference of the circle is ______.

4. The ______of the ______is 2푟.

5. The ratio of the circumference to the diameter is ______.

1 1 6. Area (circle) = Area of (______) = ∙ circumference ∙ 푟 = (2휋푟) ∙ 푟 = 휋 ∙ 푟 ∙ 푟 = ______. 2 2

Word bank

radius height base ퟐ흅풓

diameter circle rectangle 흅풓ퟐ 흅

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Exit Ticket Sample Solutions

Complete each statement using the words or algebraic expressions listed in the word bank below.

1. The length of the height of the rectangular region approximates the length of the radius of the circle. 2. The base of the rectangle approximates the length of one-half of the circumference of the circle. 3. The circumference of the circle is ퟐ훑퐫. 4. The diameter of the circle is ퟐ퐫. 5. The ratio of the circumference to the diameter is 흅. ퟏ ퟏ = = ∙ 퐜퐢퐫퐜퐮퐦퐟퐞퐫퐞퐧퐜퐞 ∙ 풓 = (ퟐ흅풓) ∙ 풓 = 흅 ∙ 풓 ∙ 풓 = 훑퐫ퟐ. 6. Area (circle) Area of (rectangle) ퟐ ퟐ

Problem Set Sample Solutions

ퟐퟐ 1. The following circles are not drawn to scale. Find the area of each circle. (Use as an approximation for 흅.) ퟕ

ퟒퟓ 퐜퐦 ퟐ

ퟑퟒퟔ. ퟓ 퐜퐦ퟐ ퟓ, ퟏퟓퟓ. ퟏ 퐟퐭ퟐ ퟏ, ퟓퟗퟏ. ퟏ 퐜퐦ퟐ

2. A circle has a diameter of ퟐퟎ inches. a. Find the exact area, and find an approximate area using 흅 ≈ ퟑ. ퟏퟒ.

If the diameter is ퟐퟎ 퐢퐧., then the radius is ퟏퟎ 퐢퐧. If 퐀 = 훑퐫ퟐ, then 퐀 = 훑 ∙ (ퟏퟎ 퐢퐧. )ퟐ or ퟏퟎퟎ훑 퐢퐧ퟐ. 퐀 ≈ (ퟏퟎퟎ ∙ ퟑ. ퟏퟒ) 퐢퐧ퟐ ≈ ퟑퟏퟒ 퐢퐧ퟐ.

b. What is the circumference of the circle using 흅 ≈ ퟑ. ퟏퟒ ?

If the diameter is ퟐퟎ 퐢퐧., then the circumference is 퐂 = 훑퐝 or 퐂 ≈ ퟑ. ퟏퟒ ∙ ퟐퟎ 퐢퐧. ≈ ퟔퟐ. ퟖ 퐢퐧.

3. A circle has a diameter of ퟏퟏ inches. a. Find the exact area and an approximate area using 흅 ≈ ퟑ. ퟏퟒ.

ퟏퟏ ퟏퟏ ퟐ ퟏퟐퟏ ퟏퟏ 퐢퐧. 퐢퐧 퐀 = 훑퐫ퟐ 퐀 = 훑 ∙ ( 퐢퐧. ) 훑 퐢퐧ퟐ. If the diameter is , then the radius is ퟐ . If , then ퟐ or ퟒ ퟏퟐퟏ 퐀 ≈ ( ∙ ퟑ. ퟏퟒ) 퐢퐧ퟐ ≈ ퟗퟒ. ퟗퟖퟓ 퐢퐧ퟐ ퟒ

b. What is the circumference of the circle using 흅 ≈ ퟑ. ퟏퟒ?

If the diameter is ퟏퟏ inches, then the circumference is 퐂 = 훑퐝 or 퐂 ≈ ퟑ. ퟏퟒ ∙ ퟏퟏ 퐢퐧. ≈ ퟑퟒ. ퟓퟒ 퐢퐧.

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4. Using the figure below, find the area of the circle.

In this circle, the diameter is the same as the length of the side of the square. The diameter is ퟏퟎ 퐜퐦; so, the radius is ퟓ 퐜퐦. 푨 = 흅풓ퟐ, so 푨 = 흅(ퟓ 퐜퐦)ퟐ = ퟐퟓ흅 퐜퐦ퟐ.

5. A path bounds a circular lawn at a park. If the inner edge of the path is ퟏퟑퟐ 퐟퐭. around, approximate the amount of ퟐퟐ area of the lawn inside the circular path. Use 흅 ≈ . ퟕ The length of the path is the same as the circumference. Find the radius from the circumference; then, find the area. 푪 = ퟐ흅풓 ퟐퟐ ퟏퟑퟐ 퐟퐭. ≈ ퟐ ∙ ∙ 풓 ퟕ ퟒퟒ ퟏퟑퟐ 퐟퐭. ≈ 풓 ퟕ ퟕ ퟕ ퟒퟒ ∙ ퟏퟑퟐ 퐟퐭. ≈ ∙ 풓 ퟒퟒ ퟒퟒ ퟕ ퟐퟏ 퐟퐭. ≈ 풓

ퟐퟐ 푨 ≈ ∙ (ퟐퟏ 퐟퐭. )ퟐ ퟕ 푨 ≈ ퟏퟑퟖퟔ 퐟퐭ퟐ

6. The area of a circle is ퟑퟔ흅 퐜퐦ퟐ. Find its circumference.

Find the radius from the area of the circle; then, use it to find the circumference.

푨 = 흅풓ퟐ ퟑퟔ흅 퐜퐦ퟐ = 흅풓ퟐ ퟏ ퟏ ∙ ퟑퟔ흅 퐜퐦ퟐ = ∙ 흅풓ퟐ 흅 흅 ퟑퟔ 퐜퐦ퟐ = 풓ퟐ ퟔ 퐜퐦 = 풓

푪 = ퟐ흅풓 푪 = ퟐ흅 ∙ ퟔ 퐜퐦 푪 = ퟏퟐ흅 퐜퐦

7. Find the ratio of the area of two circles with radii ퟑ 퐜퐦 and ퟒ 퐜퐦.

The area of the circle with radius ퟑ 퐜퐦 is ퟗ흅 퐜퐦ퟐ. The area of the circle with the radius ퟒ 퐜퐦 is ퟏퟔ흅 퐜퐦ퟐ. The ratio of the area of the two circles is ퟗ흅 ∶ ퟏퟔ흅 or ퟗ ∶ ퟏퟔ.

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8. If one circle has a diameter of ퟏퟎ 퐜퐦 and a second circle has a diameter of ퟐퟎ 퐜퐦, what is the ratio of the area of the larger circle to the area of the smaller circle?

The area of the circle with the diameter of ퟏퟎ 퐜퐦 has a radius of ퟓ 퐜퐦. The area of the circle with the diameter of ퟏퟎ 퐜퐦 is 흅 ∙ (ퟓ 퐜퐦)ퟐ, or ퟐퟓ흅 퐜퐦ퟐ. The area of the circle with the diameter of ퟐퟎ 퐜퐦 has a radius of ퟏퟎ 퐜퐦. The area of the circle with the diameter of ퟐퟎ 퐜퐦 is 흅 ∙ (ퟏퟎ 퐜퐦)ퟐ or ퟏퟎퟎ흅 퐜퐦ퟐ. The ratio of the diameters is ퟐퟎ to ퟏퟎ or ퟐ: ퟏ, while the ratio of the areas is ퟏퟎퟎ흅 풕o ퟐퟓ흅 or ퟒ: ퟏ.

9. Describe a rectangle whose perimeter is ퟏퟑퟐ 퐟퐭. and whose area is less than ퟏ 퐟퐭ퟐ. Is it possible to find a circle whose circumference is ퟏퟑퟐ 퐟퐭. and whose area is less than ퟏ 퐟퐭ퟐ? If not, provide an example or write a sentence explaining why no such circle exists.

A rectangle that has a perimeter of ퟏퟑퟐ 퐟퐭. can have a length of ퟔퟓ. ퟗퟗퟓ 퐟퐭. and a width of ퟎ . ퟎퟎퟓ 퐟퐭. The area of such a rectangle is ퟎ. ퟑퟐퟗퟗퟕퟓ 퐟퐭ퟐ, which is less than ퟏ 퐟퐭ퟐ. No, because a circle that has a circumference of ퟏퟑퟐ 퐟퐭. has a radius of approximately ퟐퟏ 퐟퐭.

푨 = 흅풓ퟐ = 흅(ퟐퟏ)ퟐ = ퟏퟑퟖퟕ. ퟗퟔ ≠ ퟏ

10. If the diameter of a circle is double the diameter of a second circle, what is the ratio of the area of the first circle to the area of the second?

If I choose a diameter of ퟐퟒ 퐜퐦 for the first circle, then the diameter of the second circle is ퟏퟐ 퐜퐦. The first circle has a radius of ퟏퟐ 퐜퐦 and an area of ퟏퟒퟒ흅 퐜퐦ퟐ. The second circle has a radius of ퟔ 퐜퐦 and an area of ퟑퟔ흅 퐜퐦ퟐ. The ratio of the area of the first circle to the second is ퟏퟒퟒ흅 to ퟑퟔ흅 , which is a ퟒ to ퟏ ratio. The ratio of the diameters is ퟐ, while the ratio of the areas is the square of ퟐ, or ퟒ.

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