Arc length & sector area
For a circle of radius r and central angle θ in radians, arc length is s = rθ and sector area is A = ½r²θ. In degrees, use the fraction of a full turn: s = (θ/360°) × 2πr and A = (θ/360°) × πr².
| Use it to | Formula |
|---|---|
| Arc length (radians) | s = rθ |
| Sector area (radians) | A = ½r²θ |
| Arc length (degrees) | s = (θ/360°) × 2πr |
| Sector area (degrees) | A = (θ/360°) × πr² |
| Sector area from arc length | A = ½rs |
| Central angle from arc length | θ = s / r (radians) |
| Chord length | 2r sin(θ/2) |
What each letter means
- r
- Radius of the circle
- θ
- Central angle between the two radii
- s
- Arc length along the curved edge
- A
- Sector area, in square units
Arc length & sector area calculator
Measure a slice of a circle. Change a value and the diagram, result, and working update together.
Your measurements
These formulas use θ in radians. The tool converts the entered degrees automatically. Angles above 180° describe major sectors.
Why it works
The idea
A central angle determines a fraction of a full turn. The same fraction applies to the circle’s circumference and its area. In radians, a full turn is 2π, which simplifies the formulas to arc length s = rθ and sector area A = r²θ/2.
Worked examples
1Sector in degrees
- r = 6 cm and θ = 120°, which is 120/360 = ⅓ of a full turn.
- Arc length = ⅓ × 2π × 6 = 4π ≈ 12.57 cm.
- Sector area = ⅓ × π × 6² = 12π ≈ 37.70 cm².
Answer: About 12.57 cm and 37.70 cm².
2Sector in radians
- r = 10 and θ = 0.8 radians.
- s = rθ = 10 × 0.8 = 8.
- A = ½r²θ = ½ × 100 × 0.8 = 40.
Answer: Arc length 8 units; sector area 40 square units.
3Find the angle from the arc length
- An arc of 15 cm lies on a circle of radius 6 cm.
- θ = s / r = 15 / 6 = 2.5 radians.
- In degrees: 2.5 × 180 / π ≈ 143.24°.
- Sector area = ½rs = ½ × 6 × 15 = 45 cm².
Answer: θ = 2.5 rad ≈ 143.24°, and the sector area is 45 cm².
Why the formulas work
- A central angle θ covers the fraction θ / 2π of a full turn (or θ / 360° in degrees).
- The arc is that fraction of the circumference: (θ / 2π) × 2πr = rθ.
- The sector is that fraction of the area: (θ / 2π) × πr² = ½r²θ.
Common mistakes
- Putting degrees into s = rθ. Convert to radians first, or use the degree versions.
- Confusing the arc (curved) with the chord (straight).
- Using the diameter instead of the radius.
- Mixing up a sector with a segment. A segment is the sector minus the triangle: ½r²(θ − sin θ).
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Keep exploring
Arc length & sector area: common questions
Wondering about the why? Start here.
What is the formula for the area of a sector?
With the angle θ in radians, sector area is A = ½r²θ. With θ in degrees, A = (θ/360) × πr². A 90° sector of a circle with radius 4 has area ¼ × 16π = 4π ≈ 12.57 square units.
Calculate a sectorHow do you find the area of a sector in radians?
Multiply half the square of the radius by the angle: A = ½r²θ. For r = 10 and θ = 0.8 radians, A = ½ × 100 × 0.8 = 40 square units. No π is needed because radians already measure the fraction of the circle.
How do you find the central angle from the arc length?
Divide the arc length by the radius: θ = s / r gives the angle in radians. Multiply by 180/π for degrees. An arc of 15 cm on a 6 cm radius gives 2.5 radians, about 143.24°.
Convert radians to degreesWhat is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, like a slice of pizza. A segment is bounded by a chord and an arc. Segment area is the sector area minus the triangle: ½r²(θ − sin θ), with θ in radians.