Arc length calculator
Enter the radius and the central angle and get the length of the arc, with the sector area, the perimeter of the sector and the circumference of the whole circle.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
-
Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
-
Instant answer
The answer appears as you type, with the most important intermediate values.
-
See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
The formula
The arc is a fraction of the full circumference 2πr: arc length = (angle ÷ 360°) × 2πr. In radians it is even simpler: arc length = r × θ.
For a radius of 10 and an angle of 60°: (60 ÷ 360) × 2π × 10 = 10.47.
Arc, sector and chord
The arc is the curved part of the edge. The sector is the slice between two radii and the arc, like a slice of pizza; its area is in the circle sector calculator. The central angle must be between 0° and 360°.
Worked examples
- A wheel: a wheel of radius 0.35 m turning 90° rolls an arc of (90 ÷ 360) × 2π × 0.35 ≈ 0.55 m.
- Round the Earth: one degree of latitude spans an arc of (1 ÷ 360) × 2π × 6,371 km ≈ 111.2 km, which is why a degree of latitude is about 111 km.
- A curved path: a garden path that follows a 120° bend of a circle with a radius of 8 m is (120 ÷ 360) × 2π × 8 ≈ 16.76 m long.
Why radians make it simple
When the angle is in radians, the arc length is just r × θ, because one radian is defined as the angle whose arc equals the radius. The calculator accepts radians, which is also the unit used in physics formulas.
Arc, chord and sector
The sector is the slice between two radii and the arc. Its area is in the circle sector calculator. For angles that do not come with a size, the angle calculator converts between degrees and radians.
Frequently asked questions
How do I find the arc length from the radius and angle?
Take the fraction of the full circle and multiply by the circumference: (angle ÷ 360°) × 2πr. In radians it is simply r × θ.
What is the difference between the arc length and the chord?
The arc follows the curve, and the chord is the straight line between its two endpoints. For 72° and radius 10, the arc is 12.57 and the chord 11.76.
How do I find the angle if I know the arc and the radius?
Divide the arc length by the radius for the angle in radians, and multiply by 180/π for degrees. An arc of 12.57 on radius 10 is 1.257 rad, which is 72°.
Can the angle be larger than 360°?
No. The central angle must be greater than 0° and at most 360°. A full 360° gives the whole circumference.