Arc Length & Sector Area Calculator
By Santaji GadepatilinUpdated October 27, 2026
An arc is part of a circle's edge, and a sector is the slice between two radii. Both depend on the radius and the angle they span. This free arc length & sector area calculator solves for whichever value you leave blank and shows the formula and steps.
Arc length (cm) vs Central angle (°) — other inputs held at your current values.
How to use the arc length & sector area calculator
Enter the radius and central angle to get the arc length, sector area and chord. Or enter the arc length and radius to find the angle. Angles can be in degrees or radians.
Formulas (θ in radians)
L = r θ A = ½ r² θ chord = 2r sin(θ ÷ 2)
To convert: radians = degrees × π ÷ 180.
Worked example
Radius 10 cm, angle 60°:
- θ = 60 × π ÷ 180 = 1.0472 rad
- Arc length = 10 × 1.0472 = 10.47 cm
- Sector area = ½ × 100 × 1.0472 = 52.36 cm²
- Chord = 2 × 10 × sin 30° = 10 cm
Arc, sector and chord for radius 10
| Angle | Radians | Arc length | Sector area | Chord |
|---|---|---|---|---|
| 30° | 0.5236 | 5.24 | 26.18 | 5.18 |
| 45° | 0.7854 | 7.85 | 39.27 | 7.65 |
| 60° | 1.0472 | 10.47 | 52.36 | 10.00 |
| 90° | 1.5708 | 15.71 | 78.54 | 14.14 |
| 120° | 2.0944 | 20.94 | 104.72 | 17.32 |
| 180° | 3.1416 | 31.42 | 157.08 | 20.00 |
| 270° | 4.7124 | 47.12 | 235.62 | 14.14 |
| 360° | 6.2832 | 62.83 | 314.16 | 0.00 |
Degrees and radians
radians = degrees × π ÷ 180
The arc-length formula L = rθ only works with the angle in radians. If you prefer degrees, use L = 2πr × (θ ÷ 360). The calculator converts automatically when you pick degrees.
Segment area
The area between the chord and the arc (a circular segment) is the sector minus the triangle:
Segment area = r² ÷ 2 × (θ − sin θ)
For r = 10 and θ = 60°, that is 50 × (1.0472 − 0.8660) = 9.06.
Finding the angle or radius
- Angle from arc length: θ = L ÷ r
- Radius from arc length: r = L ÷ θ
- Radius from a chord and height (sagitta s): r = (c² ÷ 4 + s²) ÷ (2s)
The last one is handy for measuring large curves, such as an arch or a curved wall, where you cannot reach the centre.
Practical uses
- Bending pipe or bar: the arc length is the material needed for a curved section.
- Pie charts: each slice is a sector — its angle is the share × 360°.
- Curved paths and tracks: arc length gives the distance travelled around a bend.
Frequently asked questions
What is the arc length formula?
Arc length = r × θ, with θ in radians. In degrees, arc length = 2πr × (θ ÷ 360).
How do I find the area of a sector?
Sector area = ½ × r² × θ (radians), which is the circle's area times θ ÷ 360°.
What is the difference between arc length and chord length?
The arc follows the curve; the chord is the straight line between the arc's ends. The chord is always shorter.
How do I find arc length without the angle?
If you know the chord c and radius r, the angle is θ = 2 × asin(c ÷ 2r). Then arc length = r × θ.
What is the arc length of a semicircle?
Half the circumference: π × r. For radius 10 it is about 31.42.