What this calculator does
An arc is the curved section of a circle’s edge that sits between two radii, and its length depends on just two things: how big the circle is, and how wide the angle at the centre is. The arc length formula ties these together directly, without needing to work out the full circumference first and then take a slice of it by hand.
This calculator takes a radius and a central angle, in either degrees or radians, and returns the length of that arc as the headline figure, alongside the angle converted into the other unit and the share of the full circumference it represents. It is deliberately narrow: for a sector’s area, chord and segment as well, the circle sector calculator covers the whole wedge in one place.
The formula
Convert the central angle to radians if it was entered in degrees, using θ(rad) = θ(deg) × π/180. Arc length is then s = r × θ, radius multiplied by the angle in radians. This works because a radian is defined as the angle that traces out an arc equal in length to the radius, so multiplying by the angle in radians scales that one-radius-length arc up to whatever angle you have.
| Term | Meaning |
|---|---|
| s | Arc length, the curved distance along the circle between the two ends of the angle. |
| r | Radius of the circle. |
| θ | The central angle, the angle the arc subtends at the centre of the circle. |
The inputs explained
| Field | What to enter |
|---|---|
| Radius | The radius of the circle the arc belongs to, in any consistent unit. |
| Central angle | The central angle. Switch the unit selector to match whether you are working in degrees or radians. |
| Angle unit | Choose from Degrees, Radians. |
When to use it
Cutting a curved section of trim or edging
A curved garden bed edge, a section of pipe lagging, or a strip of trim following part of a circular wall all need a length measurement along the curve, which is exactly what the arc length formula gives from a radius and the angle swept.
Working a geometry problem stated in radians
Once angles are given in radians rather than degrees, arc length is the most direct application: s = rθ needs no further conversion, which the radian unit option here handles without extra steps.
Checking a CNC or laser-cutting path
Programming a curved cut often specifies a radius and a sweep angle, and confirming the resulting arc length independently catches a keyed-in angle error before material is cut.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How arc length changes with the central angle
A fixed radius of 10, across central angles from a narrow sliver to a full half-circle.
| Central angle | Arc length | Share of the full circumference |
|---|---|---|
| 15° | 2.618 | 4.17% |
| 30° | 5.236 | 8.33% |
| 60° | 10.472 | 16.7% |
| 90° | 15.708 | 25.0% |
| 180° | 31.416 | 50.0% |
| 270° | 47.124 | 75.0% |
How arc length changes with the radius
A fixed 60° central angle, across a range of radius values.
| Radius | Arc length | Full circumference |
|---|---|---|
| 2 | 2.094 | 12.566 |
| 5 | 5.236 | 31.416 |
| 10 | 10.472 | 62.832 |
| 20 | 20.944 | 125.664 |
| 50 | 52.360 | 314.159 |
| 100 | 104.720 | 628.319 |
Questions
What is the arc length formula?
Arc length s = rθ, where r is the radius and θ is the central angle in radians. If the angle is given in degrees, convert it first with θ(rad) = θ(deg) × π/180, or use the degree option here and let the conversion happen automatically.
How do you find the length of an arc without a calculator?
Work out what fraction of a full circle the angle represents (angle ÷ 360 for degrees), then multiply that fraction by the full circumference, 2πr. That is the same result as rθ with θ in radians, just reached by a different route.
Does arc length depend on the radius or just the angle?
Both. The same angle traces a longer arc on a bigger circle, and the same circle traces a longer arc for a bigger angle. Arc length scales in direct proportion to each one individually when the other is held fixed.
What is the difference between arc length and chord length?
Arc length is the distance measured along the curve of the circle. Chord length is the straight-line distance between the same two endpoints. The chord is always shorter than the arc, except when the angle is zero, where both are zero.
For the sector’s area, chord length and segment area alongside the arc, see the circle sector calculator. For every measurement of a full circle from any one known value, see the circle calculator.