Practice
Arc Lengths and Areas of Sectors Practice
Fifty original questions on circumference, circle area, arcs, sectors, radians, wheel revolutions, and composite applications.
- Answered
- 0 / 50
- Correct
- 0
- Incorrect
- 0
- Accuracy
- 0%
Question 1
Explanation
Use C=2πr=2π(3)=6π.
- Reasoning
Distinguish radius from diameter.
Question 2
Explanation
Use C=2πr=2π(5)=10π.
- Reasoning
Distinguish radius from diameter.
Question 3
Explanation
Use C=2πr=2π(7)=14π.
- Reasoning
Distinguish radius from diameter.
Question 4
Explanation
Use C=2πr=2π(9)=18π.
- Reasoning
Distinguish radius from diameter.
Question 5
Explanation
Use C=2πr=2π(12)=24π.
- Reasoning
Distinguish radius from diameter.
Question 6
Explanation
Use A=πr²=π(2)²=4π.
- Reasoning
Square the radius, not the diameter.
Question 7
Explanation
Use A=πr²=π(4)²=16π.
- Reasoning
Square the radius, not the diameter.
Question 8
Explanation
Use A=πr²=π(6)²=36π.
- Reasoning
Square the radius, not the diameter.
Question 9
Explanation
Use A=πr²=π(8)²=64π.
- Reasoning
Square the radius, not the diameter.
Question 10
Explanation
Use A=πr²=π(11)²=121π.
- Reasoning
Square the radius, not the diameter.
Question 11
Explanation
Divide by 360: 30/360=1/12.
- Reasoning
Reduce the central-angle fraction before using a circle formula.
Question 12
Explanation
Divide by 360: 45/360=1/8.
- Reasoning
Reduce the central-angle fraction before using a circle formula.
Question 13
Explanation
Divide by 360: 60/360=1/6.
- Reasoning
Reduce the central-angle fraction before using a circle formula.
Question 14
Explanation
Divide by 360: 90/360=1/4.
- Reasoning
Reduce the central-angle fraction before using a circle formula.
Question 15
Explanation
Divide by 360: 120/360=1/3.
- Reasoning
Reduce the central-angle fraction before using a circle formula.
Question 16
Explanation
L=(a/360)(2πr)=(60/360)(2π·6)=2π.
- Reasoning
Take the angle fraction of circumference.
Question 17
Explanation
L=(a/360)(2πr)=(90/360)(2π·8)=4π.
- Reasoning
Take the angle fraction of circumference.
Question 18
Explanation
L=(a/360)(2πr)=(72/360)(2π·10)=4π.
- Reasoning
Take the angle fraction of circumference.
Question 19
Explanation
L=(a/360)(2πr)=(150/360)(2π·12)=10π.
- Reasoning
Take the angle fraction of circumference.
Question 20
Explanation
L=(a/360)(2πr)=(144/360)(2π·15)=12π.
- Reasoning
Take the angle fraction of circumference.
Question 21
Explanation
L=(a/360)(2πr)=(40/360)(2π·18)=4π.
- Reasoning
Take the angle fraction of circumference.
Question 22
Explanation
L=(a/360)(2πr)=(135/360)(2π·20)=15π.
- Reasoning
Take the angle fraction of circumference.
Question 23
Explanation
L=(a/360)(2πr)=(120/360)(2π·21)=14π.
- Reasoning
Take the angle fraction of circumference.
Question 24
Explanation
A_s=(a/360)πr²=(60/360)π(6)²=6π.
- Reasoning
Take the angle fraction of total area.
Question 25
Explanation
A_s=(a/360)πr²=(90/360)π(8)²=16π.
- Reasoning
Take the angle fraction of total area.
Question 26
Explanation
A_s=(a/360)πr²=(72/360)π(10)²=20π.
- Reasoning
Take the angle fraction of total area.
Question 27
Explanation
A_s=(a/360)πr²=(150/360)π(12)²=60π.
- Reasoning
Take the angle fraction of total area.
Question 28
Explanation
A_s=(a/360)πr²=(144/360)π(15)²=90π.
- Reasoning
Take the angle fraction of total area.
Question 29
Explanation
A_s=(a/360)πr²=(40/360)π(18)²=36π.
- Reasoning
Take the angle fraction of total area.
Question 30
Explanation
A_s=(a/360)πr²=(135/360)π(20)²=150π.
- Reasoning
Take the angle fraction of total area.
Question 31
Explanation
A_s=(a/360)πr²=(120/360)π(21)²=147π.
- Reasoning
Take the angle fraction of total area.
Question 32
Explanation
Solve 5π=(θ/360)(2π·10); this gives θ=90°.
- Reasoning
Cancel pi, then solve the proportion.
Question 33
Explanation
Solve 8π=(θ/360)(2π·12); this gives θ=120°.
- Reasoning
Cancel pi, then solve the proportion.
Question 34
Explanation
Solve 10π=(θ/360)(2π·15); this gives θ=120°.
- Reasoning
Cancel pi, then solve the proportion.
Question 35
Explanation
Solve 6π=(θ/360)(2π·18); this gives θ=60°.
- Reasoning
Cancel pi, then solve the proportion.
Question 36
Explanation
Solve 25π=(θ/360)(2π·20); this gives θ=225°.
- Reasoning
Cancel pi, then solve the proportion.
Question 37
Explanation
One revolution is 2π(0.5)=1π meters, so n=10π/(1π)=10.
- Reasoning
Divide total distance by one circumference.
Question 38
Explanation
One revolution is 2π(0.25)=0.5π meters, so n=20π/(0.5π)=40.
- Reasoning
Divide total distance by one circumference.
Question 39
Explanation
One revolution is 2π(2)=4π meters, so n=30π/(4π)=7.5.
- Reasoning
Divide total distance by one circumference.
Question 40
Explanation
One revolution is 2π(1.5)=3π meters, so n=12π/(3π)=4.
- Reasoning
Divide total distance by one circumference.
Question 41
Explanation
One third of πr² equals 48π, so r²=144 and r=12.
- Reasoning
Set the sector fraction times full area equal to the given area.
Question 42
Explanation
One eighth of circumference is 3π, so C=24π and d=24.
- Reasoning
Scale the part to the full circumference, then use C=πd.
Question 43
Explanation
Half the circumference is 8π and the straight diameter is 16.
- Reasoning
Composite perimeters include both curved and straight boundaries.
Question 44
Explanation
The removed quarter has area 25π, leaving 100π-25π=75π.
- Reasoning
Use complement fraction or subtract the sector from the whole.
Question 45
Explanation
Convert 154π feet to 1848π inches. Each revolution is 28π inches, and 1848/28=66.
- Reasoning
Convert units before dividing by circumference.
Question 46
Explanation
The full circumference is 40π, so 2πr=40π and r=20.
- Reasoning
Scale the arc to a full circle first.
Question 47
Explanation
Cancel the common angle fraction: 2πr=πr², so positive r=2.
- Reasoning
Compare the whole-circle quantities because both use the same fraction.
Question 48
Explanation
Use A_s=(1/2)r²θ=(1/2)(81)(2π/3)=27π.
- Reasoning
For radians use θ/(2π), or the half-r-squared-theta shortcut.
Question 49
Explanation
Distance is 2π(30)(80)=4800π centimeters, which is 48π meters.
- Reasoning
Calculate in the given unit, then convert once.
Question 50
Explanation
The two radii contribute 24, so the arc is 10π. Solve 10π=(θ/360)(24π), giving θ=150°.
- Reasoning
Remove the two radii from sector perimeter before solving for arc length.
Keyboard: use Tab to move, arrow keys to change answer choices, and Enter to check an answer.
Your practice summary
Use the results to decide what to review before your next attempt.
- Correct
- 0
- Incorrect
- 0
- Completed
- 50 / 50
Questions to review
No mistakes this time. Excellent work.
- Question 1CircumferenceEasy
- Question 2CircumferenceEasy
- Question 3CircumferenceEasy
- Question 4CircumferenceEasy
- Question 5CircumferenceEasy
- Question 6Circle areaEasy
- Question 7Circle areaEasy
- Question 8Circle areaEasy
- Question 9Circle areaEasy
- Question 10Circle areaEasy
- Question 11Central-angle fractionEasy
- Question 12Central-angle fractionEasy
- Question 13Central-angle fractionEasy
- Question 14Central-angle fractionEasy
- Question 15Central-angle fractionEasy
- Question 16Arc lengthMedium
- Question 17Arc lengthMedium
- Question 18Arc lengthMedium
- Question 19Arc lengthMedium
- Question 20Arc lengthMedium
- Question 21Arc lengthMedium
- Question 22Arc lengthMedium
- Question 23Arc lengthMedium
- Question 24Sector areaMedium
- Question 25Sector areaMedium
- Question 26Sector areaMedium
- Question 27Sector areaMedium
- Question 28Sector areaMedium
- Question 29Sector areaMedium
- Question 30Sector areaMedium
- Question 31Sector areaMedium
- Question 32Reverse arc lengthMedium
- Question 33Reverse arc lengthMedium
- Question 34Reverse arc lengthMedium
- Question 35Reverse arc lengthMedium
- Question 36Reverse arc lengthMedium
- Question 37Wheel revolutionsMedium
- Question 38Wheel revolutionsMedium
- Question 39Wheel revolutionsMedium
- Question 40Wheel revolutionsMedium
- Question 41Reverse sector areaHard
- Question 42Reverse arc lengthHard
- Question 43Composite perimeterHard
- Question 44Composite areaHard
- Question 45Wheel unit conversionHard
- Question 46Fraction synthesisHard
- Question 47Formula synthesisHard
- Question 48Radian sector areaHard
- Question 49Wheel distance conversionHard
- Question 50Sector perimeter synthesisHard