Arc Length and Areas of Sectors

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

Worksheets for 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.

Arc Length and Areas of Sectors

Classwork

Example 1

a. What is the length of the arc of degree that measures 60° in a circle of radius 10 cm?
b. Given the concentric circles with center 𝐴 and with 𝑚∠𝐴 = 60°, calculate the arc length intercepted by ∠𝐴 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 60°, 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 𝑥° on a circle of radius 𝑟.
e. Is the length of an arc intercepted by an angle proportional to the radius? Explain.

SECTOR: Let 𝐴𝐵 be an arc of a circle with center 𝑂 and radius 𝑟. The union of all segments 𝑂𝑃, where 𝑃 is any point of 𝐴𝐵 , is called a sector

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Exercise 1

1. The radius of the following circle is 36 cm, and the 𝑚∠𝐴𝐵𝐶 = 60°.
   a. What is the arc length of 𝐴𝐶 ?
   b. What is the radian measure of the central angle?

Example 2

a. Circle 𝑂 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 60°.
e. Circle 𝑂 has a minor arc 𝐴𝐵 with an angle measure of 60°. Sector 𝐴𝑂𝐵 has an area of 24𝜋. What is the radius of circle 𝑂?
f. Give a general formula for the area of a sector defined by an arc of angle measure 𝑥° on a circle of radius r

Exercises 2–3

2. The area of sector 𝐴𝑂𝐵 in the following image is 28𝜋 cm². Find the measurement of the central angle labeled 𝑥°.
3. In the following figure of circle 𝑂, 𝑚∠𝐴𝑂𝐶 = 108° and 𝐴𝐵 = 𝐴𝐶 = 10 cm.
   a. Find 𝑚∠𝑂𝐴𝐵.
   b. Find 𝑚𝐵𝐶
   c. Find the area of sector 𝐵𝑂𝐶

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 𝑂, let 𝐴 and 𝐵 be different points that lie on the circle but are not the endpoints of a diameter. The minor arc between 𝐴 and 𝐵 is the set containing 𝐴, 𝐵, and all points of the circle that are in the interior of ∠𝐴𝑂𝐵. The major arc is the set containing 𝐴, 𝐵, and all points of the circle that lie in the exterior of ∠𝐴𝑂𝐵.
• 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 𝐴𝐵 be an arc of a circle with center 𝑂 and radius 𝑟. The union of the segments 𝑂𝑃, where 𝑃 is any point on 𝐴𝐵 , is called a sector. 𝐴𝐵 is called the arc of the sector, and 𝑟 is called its radius.
• SEMICIRCLE: In a circle, let 𝐴 and 𝐵 be the endpoints of a diameter. A semicircle is the set containing 𝐴, 𝐵, and all points of the circle that lie in a given half-plane of the line determined by the diameter.

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