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Lesson 17: the Area of a Circle

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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). Lesson 17: The Area of a Circle 252 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G7-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 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. Lesson 17: The Area of a Circle 253 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G7-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 7•3 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 Lesson 17: The Area of a Circle 254 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G7-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 7•3 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 Lesson 17: The Area of a Circle 255 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G7-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 7•3 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. ퟐ ퟐ 퐀 = 훑 ∙ (ퟐ 퐜퐦) = ퟒ훑 퐜퐦 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. Lesson 17: The Area of a Circle 256 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G7-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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