Arc Length and Areas of Sectors
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
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
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