# Arc Length and Areas of Sectors

### New York State Common Core Math Geometry, Module 5, Lesson 9

Worksheets for Geometry, Module 5, Lesson 9

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.

Arc Length and Areas of Sectors

Classwork

Example 1

a. What is the length of the arc of degree that measures 60Β° in a circle of radius 10 cm?
b. Given the concentric circles with center π΄ and with πβ π΄ = 60Β°, calculate the arc length intercepted by β π΄ on each circle. The inner circle has a radius of 10, and each circle has a radius 10 units greater than the previous circle.
c. An arc, again of degree measure 60Β°, has an arc length of 5π cm. What is the radius of the circle on which the arc sits?
d. Give a general formula for the length of an arc of degree measure π₯Β° on a circle of radius π.
e. Is the length of an arc intercepted by an angle proportional to the radius? Explain.

SECTOR: Let π΄π΅ be an arc of a circle with center π and radius π. The union of all segments ππ, where π is any point of π΄π΅ , is called a sector

Exercise 1

1. The radius of the following circle is 36 cm, and the πβ π΄π΅πΆ = 60Β°.
a. What is the arc length of π΄πΆ ?
b. What is the radian measure of the central angle?

Example 2

a. Circle π has a radius of 10 cm. What is the area of the circle? Write the formula.
b. What is the area of half of the circle? Write and explain the formula.
c. What is the area of a quarter of the circle? Write and explain the formula.
d. Make a conjecture about how to determine the area of a sector defined by an arc measuring 60Β°.
e. Circle π has a minor arc π΄π΅ with an angle measure of 60Β°. Sector π΄ππ΅ has an area of 24π. What is the radius of circle π?
f. Give a general formula for the area of a sector defined by an arc of angle measure π₯Β° on a circle of radius r

Exercises 2β3

1. The area of sector π΄ππ΅ in the following image is 28π cm2. Find the measurement of the central angle labeled π₯Β°.
2. In the following figure of circle π, πβ π΄ππΆ = 108Β° and π΄π΅ = π΄πΆ = 10 cm.
a. Find πβ ππ΄π΅.
b. Find ππ΅πΆ
c. Find the area of sector π΅ππΆ

Lesson Summary

Relevant Vocabulary

• ARC: An arc is any of the following three figuresβ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 π, let π΄ and π΅ be different points that lie on the circle but are not the endpoints of a diameter. The minor arc between π΄ and π΅ is the set containing π΄, π΅, and all points of the circle that are in the interior of β π΄ππ΅. The major arc is the set containing π΄, π΅, and all points of the circle that lie in the exterior of β π΄ππ΅.
• 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 π΄π΅ be an arc of a circle with center π and radius π. The union of the segments ππ, where π is any point on π΄π΅ , is called a sector. π΄π΅ is called the arc of the sector, and π is called its radius.
• SEMICIRCLE: In a circle, let π΄ and π΅ be the endpoints of a diameter. A semicircle is the set containing π΄, π΅, and all points of the circle that lie in a given half-plane of the line determined by the diameter.

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