Geometry Practice Test: Arc Length, Sector Area and Radians
- Arc Length and Sector Area
- 10 questions
- 13 minutes
These 10 questions use the fraction of the circle that a central angle cuts off. Tip: arc length uses the circumference, sector area uses the area, and in radians the formulas are s = rθ and A = ½r²θ.
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Question 1
A circle has radius 9 cm. What is the length of the arc cut off by a central angle of 80°?
Correct answer: B
Arc length = 80360 · 2π(9) = 29 · 18π = 4π cm. Using the diameter 18 as the radius doubles the answer to 8π.
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Question 2
What is the area of a sector with radius 6 in. and central angle 120°?
Correct answer: C
Sector area = 120360 · π(6)2 = 13 · 36π = 12π sq in. The value 4π is the arc length, not the area.
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Question 3
What is 150° in radians?
Correct answer: A
Multiply by π180: 150 · π180 = 5π6.
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Question 4
A central angle of 2.4 radians is drawn in a circle with radius 10 m. How long is the intercepted arc?
Correct answer: D
With radians, arc length s = rθ = 10 · 2.4 = 24 m. No conversion to degrees is needed.
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Question 5
What is the area of a sector with radius 4 and central angle π4 radians?
Correct answer: B
A = 12r2θ = 12 · 16 · π4 = 2π. Forgetting the 12 gives 4π.
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Question 6
An arc of length 5π is on a circle with radius 12. What is the measure of its central angle in degrees?
Correct answer: C
θ = sr = 5π12 radians, and 5π12 · 180π = 75°.
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Question 7
A pizza with a 14-inch diameter is cut into 8 equal slices. What is the area of one slice, to the nearest tenth?
Correct answer: D
The radius is 7 in., so the whole pizza is 49π sq in. One slice is 49π8 ≈ 19.2 sq in. Using 14 as the radius gives about 77.0.
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Question 8
The minute hand of a clock is 6 inches long. How far does its tip travel in 20 minutes?
Correct answer: A
20 minutes is 13 of a full turn, and the full circle is 2π(6) = 12π in. So the tip travels 13 · 12π = 4π in.
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Question 9
A sector of a circle with radius 6 has an area of 15π. What is its central angle?
Correct answer: B
θ360 · 36π = 15π, so θ = 360 · 1536 = 150°.
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Question 10
Which statement describes an angle of 1 radian?
Correct answer: A
A radian is defined by an arc equal in length to the radius. One radian is about 57.3°, and a full turn is 2π radians.
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