Lesson 9: Arc Length and Areas of Sectors

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

Student Outcomes . When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.

Lesson Notes This lesson explores the following geometric definitions:

ARC: An arc is any of the following three figures—a minor arc, a major arc, or a semicircle. 1 LENGTH OF AN ARC: The length of an arc is the circular distance around the arc.

MINOR AND MAJOR ARC: In a circle with center 푂, let 퐴 and 퐵 be different points that lie on the circle but are not the endpoints of a diameter. The minor arc between 퐴 and 퐵 is the set containing 퐴, 퐵, and all points of the circle that are in the interior of ∠퐴푂퐵. The major arc is the set containing 퐴, 퐵, and all points of the circle that lie in the exterior of ∠퐴푂퐵.

RADIAN: A radian is the measure of the central angle of a sector of a circle with arc length of one radius length.

SECTOR: Let 퐴퐵̂ be an arc of a circle with center 푂 and radius 푟. The union of the segments 푂푃, where 푃 is any point on 퐴퐵̂ , is called a sector. 퐴퐵̂ is called the arc of the sector, and 푟 is called its radius.

SEMICIRCLE: In a circle, let 퐴 and 퐵 be the endpoints of a diameter. A semicircle is the set containing 퐴, 퐵, and all points of the circle that lie in a given half-plane of the line determined by the diameter.

Classwork Opening (2 minutes) . In Lesson 7, we studied arcs in the context of the degree measure of arcs and how those measures are determined. . Today, we examine the actual length of the arc, or arc length. Think of arc length in the following way: If we laid a piece of string along a given arc and then measured it against a ruler, this length would be the arc length.

1This definition uses the undefined term distance around a circular arc (G-CO.A.1). In Grade 4, students might use wire or string to find the length of an arc.

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This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M5-TE-1.3.0 -10.2015

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M5 GEOMETRY

Example 1 (9 minutes)

. Discuss the following exercise with a partner. Scaffolding: Prompts to help struggling Example 1 students along: a. What is the length of the arc that measures ퟔퟎ° in a circle of radius ퟏퟎ 퐜퐦? . If we can describe arc ퟏ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡 = (ퟐ흅 × ퟏퟎ) length as the length of the ퟔ string that is laid along an ퟏퟎ흅 퐀퐫퐜 퐥퐞퐧퐠퐭퐡 = arc, what is the length of MP.1 ퟑ string laid around the ퟏퟎ흅 The marked arc length is 퐜퐦. entire circle? (The ퟑ circumference, 2휋푟)

. What portion of the entire circle is the arc measure 60 1 60°? ( = ; the arc 360 6 Encourage students to articulate why their computation works. Students should be able measure tells us that the to describe that the arc length is a fraction of the entire circumference of the circle and 1 arc length is of the that fractional value is determined by the arc degree measure divided by 360°. This helps 6 them generalize the formula for calculating the arc length of a circle with arc degree length of the entire circle.) measure 푥° and radius 푟.

b. Given the concentric circles with center 푨 and with 풎∠푨 = ퟔퟎ°, calculate the arc length intercepted by ∠푨 on each circle. The inner circle has a radius of ퟏퟎ, and each circle has a radius ퟏퟎ units greater than the previous circle. ퟔퟎ ퟏퟎ흅 푨푩̅̅̅̅ = ( ) (ퟐ흅)(ퟏퟎ) = Arc length of circle with radius ퟑퟔퟎ ퟑ ퟔퟎ ퟐퟎ흅 푨푪̅̅̅̅ = ( ) (ퟐ흅)(ퟐퟎ) = Arc length of circle with radius ퟑퟔퟎ ퟑ ퟔퟎ ퟑퟎ흅 푨푫̅̅̅̅ = ( ) (ퟐ흅)(ퟑퟎ) = = ퟏퟎ흅 Arc length of circle with radius ퟑퟔퟎ ퟑ

c. An arc, again of degree measure ퟔퟎ°, has an arc length of ퟓ흅 퐜퐦. What is the radius of the circle on which the arc sits? ퟏ (ퟐ흅 × 풓) = ퟓ흅 ퟔ ퟐ흅풓 = ퟑퟎ흅 풓 = ퟏퟓ

The radius of the circle on which the arc sits is ퟏퟓ 퐜퐦.

. Notice that provided any two of the following three pieces of information—the radius, the central angle (or arc degree measure), or the arc length—we can determine the third piece of information.

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d. Give a general formula for the length of an arc of degree measure 풙° on a circle of radius 풓. MP.8 풙 = ( ) ퟐ흅풓 Arc length ퟑퟔퟎ

e. Is the length of an arc intercepted by an angle proportional to the radius? Explain. ퟐ흅풙 Yes, the arc length is a constant times the radius when 풙 is a constant angle measure, so it is ퟑퟔퟎ proportional to the radius of an arc intercepted by an angle.

Support parts (a)–(d) with these follow-up questions regarding arc lengths. Draw the corresponding figure with each question as the question is posed to the class. . From the belief that for any number between 0 and 360, there is an angle of that measure, it follows that for any length between 0 and 2휋푟, there is an arc of that length on a circle of radius 푟. . Additionally, we drew a parallel with the 180° protractor axiom (angles add) in Lesson 7 with respect to arcs. For example, if we have 퐴퐵̂ and 퐵퐶̂ as in the following figure, what can we conclude about 푚퐴퐶̂ ?

 푚퐴퐶̂ = 푚퐴퐵̂ + 푚퐵퐶̂ . We can draw the same parallel with arc lengths. With respect to the same figure, we can say arc length(퐴퐶̂ ) = arc length(퐴퐵̂ ) + arc length(퐵퐶̂ ).

. Then, given any minor arc, such as 퐴퐵̂ , what must the sum of a minor arc and its corresponding major arc (in this example, 퐴푋퐵̂) sum to?  The sum of their arc lengths is the entire circumference of the circle, or 2휋푟. . What is the possible range of lengths of any arc length? Can an arc length have a length of 0? Why or why not?  No, an arc has, by definition, two different endpoints. Hence, its arc length is always greater than zero. . Can an arc length have the length of the circumference, 2휋푟?

Students may disagree about this. Confirm that an arc length refers to a portion of a full circle. Therefore, arc lengths fall between 0 and 2휋푟; 0 < arc length < 2휋푟.

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Discussion (8 minutes) Introduce the term radian, and briefly explain its connection to the work in Example 1. Discuss what a sector is and how to find the area of a sector. 10휋 . In part (a), the arc length is . Look at part (b). Have students calculate the arc length as the central angle 3 stays the same, but the radius of the circle changes. If students write out the calculations, they see the relationship and constant of proportionality that they are trying to discover through the similarity of the circles. . What variable is determining arc length as the central angle remains constant? Why?  The radius determines the length of the arc because all circles are similar. . Is the length of an arc intercepted by an angle proportional to the radius? If so, what is the constant of proportionality? 2휋푥 휋푥  Yes, or , where 푥 is a constant angle measure in degree and 360 180 휋 the constant of proportionality is . 180 . What is the arc length if the central angle has a measure of 1°? 휋  multiplied by the length of the radius 180 휋 . Since all circles are similar, a central angle of 1° produces an arc of length multiplied by the radius. Repeat 180 that with me. 휋  Since all circles are similar, a central angle of 1° produces an arc of length multiplied by the radius. 180 . We extend our understanding of circles to include sectors. A sector can be thought of as the portion of a disk defined by an arc.

SECTOR: Let 푨푩̂ be an arc of a circle with center 푶 and radius 풓. The union of all segments 푶푷̅̅̅̅, where 푷 is any point of 푨푩̂ , is called a sector.

휋 . We can use the constant of proportionality to define a new angle measure, a radian. A radian is the 180 measure of the central angle of a sector of a circle with arc length of one radius length. Say that with me.  A radian is the measure of the central angle of a sector of a circle with arc length of one radius length.

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휋 . So, 1° = radians. What does 180° equal in radian measure? 180  휋 radians . What does 360° or a rotation through a full circle equal in radian measure?  2휋 radians . Notice, this is consistent with what we found above. You will learn more about radian measure and why it was developed in Algebra II and Calculus.

Exercise 1 (5 minutes)

Exercise 1

1. The radius of the following circle is ퟑퟔ 퐜퐦, and the 풎∠푨푩푪 = ퟔퟎ°. a. What is the arc length of 푨푪̂ ?

The degree measure of 푨푪̂ is ퟏퟐퟎ°. Then the arc length of 푨푪̂ is calculated by ퟏ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡 = (ퟐ흅 ∙ ퟑퟔ) ퟑ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡 = ퟐퟒ흅.

The arc length of 푨푪̂ is ퟐퟒ흅 퐜퐦.

b. What is the radian measure of the central angle?

퐀퐫퐜 퐥퐞퐧퐠퐭퐡 = (퐚퐧퐠퐥퐞 퐦퐞퐚퐬퐮퐫퐞 퐨퐟 퐜퐞퐧퐭퐫퐚퐥 퐚퐧퐠퐥퐞 퐢퐧 퐫퐚퐝퐢퐚퐧퐬)(퐫퐚퐝퐢퐮퐬) 퐀퐫퐜 퐥퐞퐧퐠퐭퐡 = (퐚퐧퐠퐥퐞 퐦퐞퐚퐬퐮퐫퐞 퐨퐟 퐜퐞퐧퐭퐫퐚퐥 퐚퐧퐠퐥퐞 퐢퐧 퐫퐚퐝퐢퐚퐧퐬)(ퟑퟔ) ퟐퟒ흅 = ퟑퟔ(퐚퐧퐠퐥퐞 퐦퐞퐚퐬퐮퐫퐞 퐨퐟 퐜퐞퐧퐭퐫퐚퐥 퐚퐧퐠퐥퐞 퐢퐧 퐫퐚퐝퐢퐚퐧퐬) ퟐퟒ흅 ퟐ흅 (퐚퐧퐠퐥퐞 퐦퐞퐚퐬퐮퐫퐞 퐨퐟 퐜퐞퐧퐭퐫퐚퐥 퐚퐧퐠퐥퐞 퐢퐧 퐫퐚퐝퐢퐚퐧퐬) = = ퟑퟔ ퟑ ퟐ흅 The measure of the central angle is radians. ퟑ

Example 2 (8 minutes)

Allow students to work in partners or small groups on the questions before offering Scaffolding: prompts. We calculated arc length by determining the portion of the Example 2 circumference the arc spanned. a. Circle 푶 has a radius of ퟏퟎ 퐜퐦. What is the area of the circle? Write the formula. How can we use this idea to 퐀퐫퐞퐚 = 흅(ퟏퟎ 퐜퐦)ퟐ = ퟏퟎퟎ흅 퐜퐦ퟐ find the area of a sector in terms of area of the entire

disk? (We can find the area of the sector by determining what portion of the area of the whole circle it is.)

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b. What is the area of half of the circle? Write and explain the formula.

ퟏ ퟏ ퟏퟖퟎ 퐀퐫퐞퐚 = (흅(ퟏퟎ 퐜퐦)ퟐ) = ퟓퟎ흅 퐜퐦ퟐ. ퟏퟎ 퐜퐦 is the radius of the circle, and = , which is the fraction of ퟐ ퟐ ퟑퟔퟎ the circle.

c. What is the area of a quarter of the circle? Write and explain the formula.

ퟏ ퟏ ퟗퟎ 퐀퐫퐞퐚 = (흅(ퟏퟎ 퐜퐦)ퟐ) = ퟐퟓ흅 퐜퐦ퟐ. ퟏퟎ 퐜퐦 is the radius of the circle, and = , which is the fraction of ퟒ ퟒ ퟑퟔퟎ the circle.

d. Make a conjecture about how to determine the area of a sector defined by an arc measuring ퟔퟎ°. ퟔퟎ ퟏ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푨푶푩) = (흅(ퟏퟎ 퐜퐦)ퟐ) = (흅(ퟏퟎ 퐜퐦)ퟐ) ퟑퟔퟎ ퟔ ; the area of the circle times the arc measure divided by ퟑퟔퟎ MP.7 ퟓퟎ흅 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푨푶푩) = 퐜퐦ퟐ ퟑ ퟓퟎ흅 The area of the sector 푨푶푩 is 퐜퐦ퟐ. ퟑ

Again, as with Example 1, part (a), encourage students to articulate why the computation works.

e. Circle 푶 has a minor arc 푨푩̂ with an angle measure of ퟔퟎ°. Sector 푨푶푩 has an area of ퟐퟒ흅. What is the radius of circle 푶? ퟏ ퟐퟒ흅 = (흅풓ퟐ) ퟔ ퟏퟒퟒ흅 = (흅풓ퟐ) 풓 = ퟏퟐ

The radius has a length of ퟏퟐ units.

f. Give a general formula for the area of a sector defined by an arc of angle measure 풙° on a circle of radius 풓. 풙 퐀퐫퐞퐚 퐨퐟 퐬퐞퐜퐭퐨퐫 = ( ) 흅풓ퟐ ퟑퟔퟎ

Exercises 2–3 (7 minutes)

Exercises 2–3

2. The area of sector 푨푶푩 in the following image is ퟐퟖ흅 퐜퐦ퟐ. Find the measurement of the central angle labeled 풙°. 풙 ퟐퟖ흅 = (흅(ퟏퟐ)ퟐ) ퟑퟔퟎ 풙 = ퟕퟎ

The central angle has a measurement of ퟕퟎ°.

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3. In the following figure of circle 푶, 풎∠푨푶푪 = ퟏퟎퟖ° and 푨푩̂ = 푨푪̂ = ퟏퟎ 퐜퐦. a. Find 풎∠푶푨푩. ퟑퟔ°

b. Find 풎푩푪̂ .

ퟏퟒퟒ°

c. Find the area of sector 푩푶푪. ퟏퟒퟒ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푩푶푪) = (흅(ퟓ. ퟑퟎퟓ)ퟐ) ퟑퟔퟎ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푩푶푪) ≈ ퟑퟓ. ퟑퟕ

The area of sector 푩푶푪 is ퟑퟓ. ퟑퟕ 퐜퐦ퟐ.

Closing (1 minute) Present the following questions to the entire class, and have a discussion. . What is the formula to find the arc length of a circle provided the radius 푟 and an arc of angle measure 푥°? 푥  Arc length = ( ) (2휋푟) 360 . What is the formula to find the area of a sector of a circle provided the radius 푟 and an arc of angle measure 푥°? 푥  Area of sector = ( ) (휋푟2) 360 . What is a radian?  A radian is the measure of the central angle of a sector of a circle with arc length of one radius length.

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Lesson Summary

Relevant Vocabulary . ARC: An arc is any of the following three figures—a minor arc, a major arc, or a semicircle. . LENGTH OF AN ARC: The length of an arc is the circular distance around the arc. . MINOR AND MAJOR ARC: In a circle with center 푶, let 푨 and 푩 be different points that lie on the circle but are not the endpoints of a diameter. The minor arc between 푨 and 푩 is the set containing 푨, 푩, and all points of the circle that are in the interior of ∠푨푶푩. The major arc is the set containing 푨, 푩, and all points of the circle that lie in the exterior of ∠푨푶푩. . RADIAN: A radian is the measure of the central angle of a sector of a circle with arc length of one radius length.

. SECTOR: Let 푨푩̂ be an arc of a circle with center 푶 and radius 풓. The union of the segments 푶푷, where 푷 is any point on 푨푩̂ , is called a sector. 푨푩̂ is called the arc of the sector, and 풓 is called its radius. . SEMICIRCLE: In a circle, let 푨 and 푩 be the endpoints of a diameter. A semicircle is the set containing 푨, 푩, and all points of the circle that lie in a given half-plane of the line determined by the diameter.

Exit Ticket (5 minutes)

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Name Date

Lesson 9: Arc Length and Areas of Sectors

Exit Ticket

1. Find the arc length of 푃푄푅̂.

2. Find the area of sector 푃푂푅.

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Exit Ticket Sample Solutions

1. Find the arc length of 푷푸푹̂ . ퟏퟔퟐ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푷푹̂ ) = (ퟐ흅)(ퟏퟓ) ퟑퟔퟎ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푷푹̂ ) = ퟏퟑ. ퟓ흅 퐂퐢퐫퐜퐮퐦퐟퐞퐫퐞퐧퐜퐞(퐜퐢퐫퐜퐥퐞 푶) = ퟑퟎ흅

The arc length of 푷푸푹̂ is (ퟑퟎ흅 − ퟏퟑ. ퟓ흅) 퐜퐦 or ퟏퟔ. ퟓ흅 퐜퐦.

2. Find the area of sector 푷푶푹. ퟏퟔퟐ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푷푶푹) = (흅(ퟏퟓ)ퟐ) ퟑퟔퟎ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푷푶푹) = ퟏퟎퟏ. ퟐퟓ흅

The area of sector 푷푶푹 is ퟏퟎퟏ. ퟐퟓ흅 퐜퐦ퟐ.

Problem Set Sample Solutions

1. 푷 and 푸 are points on the circle of radius ퟓ 퐜퐦, and the measure of 푷푸̂ is ퟕퟐ°. Find, to one decimal place, each of the following. a. The length of 푷푸̂ ퟕퟐ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푷푸̂ ) = (ퟐ흅 × ퟓ) ퟑퟔퟎ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푷푸̂ ) = ퟐ흅

The arc length of 푷푸̂ is ퟐ흅 퐜퐦 or approximately ퟔ. ퟑ 퐜퐦.

b. The ratio of the arc length to the radius of the circle 흅 ퟐ흅 ∙ ퟕퟐ = radians ퟏퟖퟎ ퟓ

c. The length of chord 푷푸̅̅̅̅ 푥

The length of 푷푸̅̅̅̅ is twice the value of 풙 in △ 푶푸푹. 풙 = ퟓ 퐬퐢퐧 ퟑퟔ° 푷푸 = ퟐ풙 = ퟏퟎ 퐬퐢퐧 ퟑퟔ°

Chord 푷푸̅̅̅̅ has a length of ퟏퟎ 퐬퐢퐧 ퟑퟔ 퐜퐦 or approximately ퟓ. ퟗ 퐜퐦.

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d. The distance of the chord 푷푸̅̅̅̅ from the center of the circle

The distance of chord 푷푸̅̅̅̅ from the center of the circle is labeled as 풚 in △ 푶푸푹. 풚 = ퟓ 퐜퐨퐬 ퟑퟔ°

The distance of chord 푷푸̅̅̅̅ from the center of the circle is ퟓ 퐜퐨퐬 ퟑퟔ 퐜퐦, or approximately ퟒ 퐜퐦.

e. The perimeter of sector 푷푶푸

퐏퐞퐫퐢퐦퐞퐭퐞퐫(퐬퐞퐜퐭퐨퐫 푷푶푸) = ퟓ + ퟓ + ퟐ흅 퐏퐞퐫퐢퐦퐞퐭퐞퐫(퐬퐞퐜퐭퐨퐫 푷푶푸) = ퟏퟎ + ퟐ흅 The perimeter of sector 푷푶푸 is (ퟏퟎ + ퟐ흅) 퐜퐦, or approximately ퟏퟔ. ퟑ 퐜퐦.

f. The area of the wedge between the chord 푷푸̅̅̅̅ and 푷푸̂

퐀퐫퐞퐚(퐰퐞퐝퐠퐞) = 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푷푶푸) − 퐀퐫퐞퐚(△ 푷푶푸) ퟏ 퐀퐫퐞퐚(△ 푷푶푸) = (ퟏퟎ 퐬퐢퐧 ퟑퟔ)(ퟓ 퐜퐨퐬 ퟑퟔ) ퟐ ퟕퟐ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푷푶푹) = (흅(ퟓ)ퟐ) ퟑퟔퟎ ퟕퟐ ퟏ 퐀퐫퐞퐚(퐰퐞퐝퐠퐞) = (흅(ퟓ)ퟐ) − (ퟏퟎ 퐬퐢퐧 ퟑퟔ)(ퟓ 퐜퐨퐬 ퟑퟔ) ퟑퟔퟎ ퟐ The area of wedge between chord 푷푸̅̅̅̅ and the arc 푷푸 is approximately ퟑ. ퟖ 퐜퐦ퟐ.

g. The perimeter of this wedge

퐏퐞퐫퐢퐦퐞퐭퐞퐫(퐰퐞퐝퐠퐞) = ퟐ흅 + ퟏퟎ 퐬퐢퐧 ퟑퟔ The perimeter of the wedge is approximately ퟏퟐ. ퟐ 퐜퐦.

2. What is the radius of a circle if the length of a ퟒퟓ° arc is ퟗ흅? ퟒퟓ ퟗ흅 = (ퟐ흅풓) ퟑퟔퟎ 풓 = ퟑퟔ

The radius of the circle is ퟑퟔ.

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3. 푨푩̂ and 푪푫̂ both have an angle measure of ퟑퟎ°, but their arc lengths are not the same. 푶푩 = ퟒ and 푩푫 = ퟐ. a. What are the arc lengths of 푨푩̂ and 푪푫̂ ? ퟑퟎ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푨푩̂ ) = (ퟐ흅)(ퟒ) ퟑퟔퟎ ퟐ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푨푩̂ ) = 흅 ퟑ ퟐ The arc length of 푨푩̂ is 흅. ퟑ ퟑퟎ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푪푫̂ ) = (ퟐ흅)(ퟔ) ퟑퟔퟎ 퐀퐫퐜 퐥퐞퐧퐠퐭퐡(푪푫̂ ) = 흅

The arc length of 푪푫̂ is 흅.

b. What is the ratio of the arc length to the radius for both of these arcs? Explain. ퟑퟎ흅 흅 = 퐫퐚퐝퐢퐚퐧퐬. The angle is constant, so the ratio of arc length to radius will be the angle measure, ퟑퟎ°, ퟏퟖퟎ ퟔ 흅 multiplied by . ퟏퟖퟎ

c. What are the areas of the sectors 푨푶푩 and 푪푶푫? ퟑퟎ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푨푶푩) = (흅(ퟒ)ퟐ) ퟑퟔퟎ ퟒ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푨푶푩) = 흅 ퟑ ퟒ The area of the sector 푨푶푩 is 흅. ퟑ ퟑퟎ 퐀퐫퐞퐚(퐬퐞퐜퐭퐨퐫 푪푶푫) = (흅(ퟔ)ퟐ) ퟑퟔퟎ The area of the sector 푪푶푫 is ퟑ흅.

4. In the circles shown, find the value of 풙. Figures are not drawn to scale. a. The circles have central angles of equal measure. b.

푥

흅 흅 ퟐ흅 풙 = 풙 = (ퟒ) ( ) = ퟔ ퟔ ퟑ ퟐ흅 흅 radians radians ퟑ ퟔ

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c. d.

ퟏퟖ ퟐ흅 풙 = 풙 = ퟓ ퟒퟓ

5. The concentric circles all have center 푨. The measure of the central angle is ퟒퟓ°. The arc lengths are given. a. Find the radius of each circle. 흅 ퟒퟓ흅 Radius of inner circle: = 풓, 풓 = ퟐ ퟐ ퟏퟖퟎ ퟓ흅 ퟒퟓ흅 Radius of middle circle: = 풓, 풓 = ퟓ ퟒ ퟏퟖퟎ ퟗ흅 ퟒퟓ흅 Radius of outer circle: = 풓, 풓 = ퟗ ퟒ ퟏퟖퟎ

b. Determine the ratio of the arc length to the radius of each circle, and interpret its meaning. 흅 is the ratio of the arc length to the radius of each circle. It is ퟒ the measure of the central angle in radians.

6. In the figure, if the length of 푷푸̂ is ퟏퟎ 퐜퐦, find the length of 푸푹̂ . ퟏ ퟏ Since ퟔ° is of ퟗퟎ°, then the arc length of 푸푹̂ is of ퟏퟎ 퐜퐦; the arc length of ퟏퟓ ퟏퟓ ퟐ 푸푹̂ is 퐜퐦. ퟑ

7. Find, to one decimal place, the areas of the shaded regions. a. 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = 퐀퐫퐞퐚 퐨퐟 퐬퐞퐜퐭퐨퐫 − 퐀퐫퐞퐚 퐨퐟 퐓퐫퐢퐚퐧퐠퐥퐞 ퟏ (or (퐀퐫퐞퐚 퐨퐟 퐜퐢퐫퐜퐥퐞) − 퐀퐫퐞퐚 퐨퐟 퐭퐫퐢퐚퐧퐠퐥퐞) ퟒ ퟗퟎ ퟏ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = (흅(ퟓ)ퟐ) − (ퟓ)(ퟓ) ퟑퟔퟎ ퟐ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = ퟔ. ퟐퟓ흅 − ퟏퟐ. ퟓ

The shaded area is approximately ퟕ. ퟏퟑ.

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b. The following circle has a radius of ퟐ.

ퟑ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = (퐀퐫퐞퐚 퐨퐟 퐜퐢퐫퐜퐥퐞) + 퐀퐫퐞퐚 퐨퐟 퐭퐫퐢퐚퐧퐠퐥퐞 ퟒ Note: The triangle is a ퟒퟓ°–ퟒퟓ°–ퟗퟓ° triangle with legs of length ퟐ (the legs are comprised by the radii, like the triangle in the previous question). ퟑ ퟏ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = (흅(ퟐ)ퟐ) + (ퟐ)(ퟐ) ퟒ ퟐ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = ퟑ흅 + ퟐ The shaded area is approximately ퟏퟏ. ퟒ.

퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = (퐀퐫퐞퐚 퐨퐟 ퟐ 퐬퐞퐜퐭퐨퐫퐬) + (퐀퐫퐞퐚 퐨퐟 ퟐ 퐭퐫퐢퐚퐧퐠퐥퐞퐬)

ퟗퟖ ퟒퟗ√ퟑ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = ퟐ ( 흅) + ퟒ ( ) ퟑ ퟐ ퟏퟗퟔ 퐒퐡퐚퐝퐞퐝 퐀퐫퐞퐚 = 흅 + ퟗퟖ√ퟑ ퟑ The shaded area is approximately ퟑퟕퟒ. ퟗퟗ.

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