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Arcs and Sectors - Sample Math Practice Problems

The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. In the main program, all problems are automatically graded and the difficulty adapts dynamically based on performance. Answers to these sample questions appear at the bottom of the page. This page does not grade your responses.

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Complexity=1, Mode=arc

Find the length of the arc in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

1.  
The radius of the circle is 3 in.
Length =
2.  
The radius of the circle is 8 km.
Length =

Complexity=1, Mode=sect

Find the area of the sector in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

1.  
The radius of the circle is 4 cm.
Area =
2.  
The radius of the circle is 2 in.
Area =

Complexity=2, Mode=arc

Find the length of the highlighted arc (red) in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

1.  
The radius of the circle is 10 ft.
Length =
2.  
The radius of the circle is 10 mm.
Length =

Complexity=2, Mode=sect

Find the area of the sector in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

1.  
The radius of the circle is 2 m.
Area =
2.  
The radius of the circle is 5 cm.
Area =

Complexity=3, Mode=ang

Solve.

1.   An arc has a length of 20/3 π cm and a radius of 6 cm.
Calculate the angle of the arc. °
2.   What is the measure of the central angle of a sector
if its area is 75/4 π ft2 and its radius is 5 ft? °

Answers

Complexity=1, Mode=arc

Find the length of the arc in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

#ProblemCorrect AnswerYour Answer
1
The radius of the circle is 3 in.
 Length =
Solution
Arc length = central angle
360°		•circumference
=
36
360
•2π(3 in)
=
1
10
•6π in
=
6π
10
in
Answer is
3
5
π  in.
#ProblemCorrect AnswerYour Answer
2
The radius of the circle is 8 km.
 Length =
Solution
Arc length = central angle
360°		•circumference
=
54
360
•2π(8 km)
=
3
20
•16π km
=
3 • 16π
20
km
Answer is
12
5
π  km.

Complexity=1, Mode=sect

Find the area of the sector in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

#ProblemCorrect AnswerYour Answer
1
The radius of the circle is 4 cm.
 Area =
Solution
Sector area = central angle
360°		•circle area
=
63
360
•π(4 cm)2
=
7
40
•16π cm2
=
7 • 16π
40
cm2
Answer is
14
5
π  cm2.
#ProblemCorrect AnswerYour Answer
2
The radius of the circle is 2 in.
 Area =
Solution
Sector area = central angle
360°		•circle area
=
320
360
•π(2 in)2
=
8
9
•4π in2
=
8 • 4π
9
in2
Answer is
32
9
π  in2.

Complexity=2, Mode=arc

Find the length of the highlighted arc (red) in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

#ProblemCorrect AnswerYour Answer
1
The radius of the circle is 10 ft.
 Length =
Solution
Arc length = central angle
360°		•circumference
 
= 360° - 297°
360°		•2π(10 ft)
 
=
63
360
•2π(10 ft)
=
7
40
•20π ft
=
7 • 20π
40
ft
Answer is
7
2
π  ft.
#ProblemCorrect AnswerYour Answer
2
The radius of the circle is 10 mm.
 Length =
Solution
Arc length = central angle
360°		•circumference
 
= 360° - 306°
360°		•2π(10 mm)
 
=
54
360
•2π(10 mm)
=
3
20
•20π mm
=
3 • 20π
20
mm
Answer is3π  mm.

Complexity=2, Mode=sect

Find the area of the sector in terms of pi.
Type "pi" in for π. Example: "7π m2" as "7pi sq m".

#ProblemCorrect AnswerYour Answer
1
The radius of the circle is 2 m.
 Area =
Solution
Sector area = central angle
360°		•circle area
 
= 360° - 160°
360°		•π(2 m)2
 
=
200
360
•π(2 m)2
=
5
9
•4π m2
=
5 • 4π
9
m2
Answer is
20
9
π  m2.
#ProblemCorrect AnswerYour Answer
2
The radius of the circle is 5 cm.
 Area =
Solution
Sector area = central angle
360°		•circle area
 
= 360° - 40°
360°		•π(5 cm)2
 
=
320
360
•π(5 cm)2
=
8
9
•25π cm2
=
8 • 25π
9
cm2
Answer is
200
9
π  cm2.

Complexity=3, Mode=ang

Solve.

#ProblemCorrect AnswerYour Answer
1An arc has a length of 20/3 π cm and a radius of 6 cm.
Calculate the angle of the arc.		 °
Solution
Arc length = central angle
360°		•Circumference

central angle = Arc length • 360°
2πr
= 20/3 π cm • 360°
12π cm
The answer is 200°.
#ProblemCorrect AnswerYour Answer
2What is the measure of the central angle of a sector
if its area is 75/4 π ft2 and its radius is 5 ft?		 °
Solution
Asector = central angle
360°		•Acircle

central angle = Asector • 360°
πr2
= 75/4 π ft2 • 360°
25π ft2
The answer is 270°.
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