Circle sectors and angle fractions
Use a fraction of a full turn to calculate an arc length and sector area.
Before you begin
- Circle circumference and area
- Fractions and degrees
Connect central angle to a full circle
A sector is the region between two radii and the arc joining them. A central angle of θ degrees covers θ/360 of a full circle. Its arc is the same fraction of the full circumference, and its area is the same fraction of the circle's area. These formulae use degrees; radian formulae have a different appearance.
Separate the arc from the perimeter
An arc is curved boundary only. The perimeter of a sector includes the arc and both radii. Keep exact multiples of π until the final step if a decimal is requested. Arc length uses ordinary length units, while sector area uses squared units.
Worked example
Find the area of a sector with radius 6 cm and central angle 60°.
Show the worked solution
- The sector covers 60/360 = 1/6 of the circle.
- The full circle area is π × 6² = 36π cm².
- Take one-sixth of that area.
6π cm².
Try it yourself
A 90° sector has radius 4 cm. What is its arc length?
Common mistakes
- Use the radius, not the diameter, in these formulae.
What can you explain now?
A 120° sector has radius 9 cm. Find its arc length and perimeter exactly.
Compare with the explanation
Arc 6π cm; perimeter (18 + 6π) cm.
The arc is one-third of 18π. Add two radii, totalling 18 cm, to get the complete perimeter.
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗