How arc length is calculated
An arc is a fraction of the circumference, set by the central angle.
L = r × θ (radians) · L = 2πr × θ ÷ 360 (degrees)
- Convert the angle to radians: θ = degrees × π ÷ 180.
- Multiply by the radius: L = r × θ.
- From a chord: θ = 2 × asin(c ÷ 2r), then L = r × θ.
- Chord length is 2r × sin(θ ÷ 2). Sector area is ½r²θ, and segment area is ½r²(θ − sin θ).
- To find the radius, r = L ÷ θ. To find the angle, θ = L ÷ r.
Worked example — Radius 10 cm and angle 60°: θ = π/3 = 1.0472 rad, so L = 10 × 1.0472 = 10.4720 cm. The chord is 10 cm and the sector area is 52.3599 cm².
Common arcs
| Radius | Angle | Arc length |
|---|---|---|
| 10 cm | 60 ° | 10.4720 cm |
| 5 cm | 90 ° | 7.8540 cm |
| 12 cm | 45 ° | 9.4248 cm |
| 1 cm | 180 ° | 3.1416 cm |
| 8 cm | 120 ° | 16.7552 cm |
| 20 cm | 30 ° | 10.4720 cm |
Swipe the table sideways to see the rest.
Radius in cm and angle in degrees. Each row is worked out by this calculator.
Things to consider
- Radians make the formula simple: L = rθ. Use degrees only if you also divide by 360.
- An arc is always longer than its chord.
- For a curved path or garden edge, measure the straight chord and the radius, then use chord mode.
- A 180° arc is half the circumference, and 90° is a quarter.
Frequently asked questions
What is the formula for arc length?
In radians, L = r × θ. In degrees, L = 2πr × θ ÷ 360. A 10 cm radius and 60° give 10.4720 cm.
How do I convert degrees to radians?
Multiply by π and divide by 180. 60° is π/3, about 1.0472 rad. 180° is π rad.
How do I find the arc length from a chord?
Find the angle with θ = 2 × asin(c ÷ 2r), then L = r × θ. A 10 cm chord on a 10 cm radius gives 60° and an arc of 10.4720 cm.
What is the difference between arc length and chord length?
The arc follows the curve. The chord is the straight line between its ends, so it is always shorter.
How do I find the area of a sector?
Sector area = ½ × r² × θ in radians. A 10 cm radius and 60° give 52.3599 cm². For the full circle, see the Circumference Calculator for the related measurements.
What is a central angle?
The angle at the centre of the circle between the two radii that meet the ends of the arc.
Results are estimates for general information only, not financial, medical or professional advice.