OneTaskBy Tim King · Updated
Arc Length Calculator
Sector Area and Chord Length From a Radius and Angle, Working Shown
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How the Arc Length and Sector Area Calculator works
This arc length calculator takes the radius of a circle and the central angle of a slice and returns the arc length, the sector area, the straight chord across the ends of the arc and the area of the segment between chord and arc. Enter the angle in degrees or radians; the working is shown line by line.
An arc is a piece of a circle's edge. The slice of pie it bounds, from the centre out to the arc, is a sector. The straight line joining the two ends of the arc is a chord, and the sliver between the chord and the arc is a segment. All four depend only on the radius and the angle at the centre.
How the arc length calculator works
The formula for arc length is s = rθ, with θ in radians, which is why the calculator converts degrees first by multiplying by π ÷ 180. Sector area is ½r²θ, the chord is 2r sin(θ ÷ 2), and the segment area is the sector minus the triangle under the chord: ½r²(θ − sin θ).
First example. A radius of 10 and an angle of 90° gives θ = 1.5708 rad, an arc of 15.71, a sector area of 78.54 and a chord of 14.14.
Second example. A 2 m pendulum swinging through 30° travels 2 × 0.5236 ≈ 1.05 m along its arc on each swing.
How many degrees of arc does a 50 cm curve on a 1 m radius cover? θ = 0.5 ÷ 1 = 0.5 rad, which is 0.5 × 180 ÷ π ≈ 28.6°.
Arc length, sector area and chord for a radius of 10 at every 15°
| Angle | Radians | Arc length | Sector area | Chord |
|---|---|---|---|---|
| 15° | 0.2618 | 2.62 | 13.09 | 2.61 |
| 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 |
| 75° | 1.3090 | 13.09 | 65.45 | 12.18 |
| 90° | 1.5708 | 15.71 | 78.54 | 14.14 |
| 105° | 1.8326 | 18.33 | 91.63 | 15.87 |
| 120° | 2.0944 | 20.94 | 104.72 | 17.32 |
| 135° | 2.3562 | 23.56 | 117.81 | 18.48 |
| 150° | 2.6180 | 26.18 | 130.90 | 19.32 |
| 165° | 2.8798 | 28.80 | 143.99 | 19.83 |
| 180° | 3.1416 | 31.42 | 157.08 | 20.00 |
| 195° | 3.4034 | 34.03 | 170.17 | 19.83 |
| 210° | 3.6652 | 36.65 | 183.26 | 19.32 |
| 225° | 3.9270 | 39.27 | 196.35 | 18.48 |
| 240° | 4.1888 | 41.89 | 209.44 | 17.32 |
| 255° | 4.4506 | 44.51 | 222.53 | 15.87 |
| 270° | 4.7124 | 47.12 | 235.62 | 14.14 |
| 285° | 4.9742 | 49.74 | 248.71 | 12.18 |
| 300° | 5.2360 | 52.36 | 261.80 | 10.00 |
| 315° | 5.4978 | 54.98 | 274.89 | 7.65 |
| 330° | 5.7596 | 57.60 | 287.98 | 5.18 |
| 345° | 6.0214 | 60.21 | 301.07 | 2.61 |
| 360° | 6.2832 | 62.83 | 314.16 | 0.00 |
Tips
- Radians make it simple. In radians the arc length is just radius times angle.
- A full turn is 2π. At 360° the arc is the whole circumference and the sector the whole area.
- Chord or arc? Measure along the curve for the arc, straight across for the chord.
FAQ
How do you find arc length without the angle?
If you know the chord c and the radius r, find the angle first: θ = 2 × arcsin(c ÷ 2r), in radians, then multiply by r. If you know the sector area instead, θ = 2A ÷ r². Either way the angle is the missing link, and once you have it the arc length follows directly.
What is the difference between a sector and a segment?
A sector is the slice bounded by two radii and the arc, like a slice of pizza. A segment is only the part between the arc and the chord, the crust end of that slice with the triangle removed. The calculator gives both so you can see how much area the triangle accounts for.
How do you convert degrees to radians?
Multiply the angle in degrees by π and divide by 180. So 90° is π ÷ 2, about 1.5708 radians, and 180° is π. To go the other way, multiply radians by 180 and divide by π. The arc length formula s = rθ only works with θ in radians.
Where is arc length used in real life?
Engineers use it to cut curved pipes and rails, road designers to set out bends, and makers to work out the length of edging around a curved bed or table. It also gives the distance travelled by a point on a turning wheel or the swing of a pendulum.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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