An arc is a fraction of the circumference: A sector with angle θ out of a full circle (360°) covers a fraction θ/360 of the full circumference.
Interactive: change the sector angle to see how arc length changes.
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Worked example — arc length
Find the arc length of a sector with radius 9 cm and angle 80°.
Step by step
- Apply the formula.
- Evaluate.
Final answer
Arc length = 4π ≈ 12.6 cm.
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Worked example — find the radius
A sector has arc length 15 cm and angle 60°.
Find its radius.
Step by step
- Write the formula and substitute.
- Solve for r.
Final answer
r ≈ 14.3 cm.
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Perimeter includes two radii: The perimeter of a sector is the arc length PLUS the two straight edges (the radii) that border it.
Worked example — perimeter
Find the perimeter of a sector with radius 12 cm and angle 120°.
Step by step
- Arc length.
- Perimeter = arc + 2 radii.
Final answer
Perimeter ≈ 49.1 cm.
Worked example — sprinkler coverage
A garden sprinkler rotates through 140° and reaches 8 m.
Find the total length of the arc it covers.
Step by step
- Apply the arc length formula.
Final answer
Arc length ≈ 19.5 m.
IB-style question — a cone from a sector
A party hat is made by rolling a sector of card into a cone. The sector has radius 18 cm (this becomes the slant height), and the hat's circular base has radius 5 cm.
(a) Find the angle of the sector.
(b) Find the arc length of the sector.
Step by step
- When the sector rolls up, its arc becomes the base circle. So the sector's arc length equals the base circumference.
- (a) The arc length of a sector is (θ/360) × the full circumference of a circle of radius 18. Set it equal to 10π and solve for θ.
- (b) The arc length is the 10π we already used.
Final answer
(a) θ = 100°. (b) arc ≈ 31.4 cm.
The flat sector (radius = slant 18 cm, angle 100°) folds into the cone — the sector's arc becomes the base circle (radius 5 cm), so arc length = 2πr.
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