Find the arc length of a circle from its radius and central angle — or work backwards to find the radius or angle. Enter any two of radius, central angle, and arc length, and the calculator also returns the sector area, chord length, and diameter.
The arc length calculator finds the length of an arc — a slice of a circle's circumference — from the circle's radius and the central angle that the arc spans. Enter any two of the radius, the central angle, and the arc length, and the calculator returns the missing one, along with the sector area, the chord length, and the diameter. It works in both degrees and radians, so whether you are doing geometry homework, laying out a curved path, or designing a gear, this tool gives you every measurement of the arc in one step.
An arc is simply part of the way around a circle. Because a full circle's circumference is 2πr and an arc is a fixed fraction of that circumference, the length of any arc follows directly from the radius and the angle it subtends at the center. This calculator applies that relationship for you and shows the working, so you can both get the answer and understand where it comes from.
Arc length is the distance along a curved line that forms part of the circumference of a circle. If you imagine walking around the edge of a circular track but stopping partway, the distance you covered is the arc length. It is measured in ordinary units of length — centimeters, meters, inches — not in degrees, because it is a real distance along the curve, not an angle.
Every arc is defined by two things: the radius of the circle it belongs to, and the central angle — the angle at the center of the circle between the two radii that meet the ends of the arc. A larger radius or a larger central angle both make the arc longer. When the central angle is a full turn (360° or 2π radians), the arc length equals the whole circumference.
The core formula for arc length is beautifully simple when the angle is measured in radians:
L = r × θ (θ in radians)
Here L is the arc length, r is the radius, and θ is the central angle in radians. If your angle is in degrees, you either convert it to radians first (multiply by π/180) or use the equivalent degree formula:
L = 2 × π × r × (θ° ÷ 360)
Both give the same result — the second is just the fraction of the full circumference (2πr) that the arc represents. This calculator accepts either unit; pick degrees or radians from the dropdown and it handles the conversion automatically.
Suppose a circle has a radius of 15 cm and you want the length of an arc that spans a central angle of 45°. First convert the angle: 45° = π/4 ≈ 0.7854 radians. Then:
So the arc is about 11.78 cm long and the pie-slice sector it bounds has an area of about 88.36 cm². Because the calculator works in every direction, you could instead enter the radius and the arc length to find the angle, or the angle and the arc length to find the radius.
From any two of the three main inputs, the calculator works out a complete description of the arc and its sector:
These three quantities describe different aspects of the same slice of a circle, and it helps to keep them straight. The arc length is the curved distance around the outside of the slice. The sector area is the flat area of the whole slice — the region bounded by the arc and the two radii, like a slice of pie. The chord is the straight line connecting the two endpoints of the arc, "cutting across" instead of following the curve. For the same arc, the chord is always shorter than the arc length, because a straight line is the shortest path between two points. As the central angle gets smaller, the chord and the arc length get closer and closer in value — which is the geometric idea behind approximating curves with straight segments.
Angles can be measured in two units, and understanding both makes the arc length formula click. Degrees divide a full circle into 360 parts and are the everyday unit. Radians measure an angle by the arc length it produces on a circle of radius 1 — which is exactly why the formula L = rθ is so clean in radians. A full circle is 2π radians = 360°, so 1 radian ≈ 57.2958°, and to convert you multiply degrees by π/180 or radians by 180/π. The calculator lets you work in whichever unit your problem uses, but internally it always converts to radians to apply L = rθ.
Because it accepts any two of the three, you can use it as an arc length calculator, a central angle calculator, or a radius calculator, all in one.
Arc length is the bridge between how far something spins and how far it actually travels. When a wheel of radius r turns through an angle θ (in radians), a point on its rim moves a distance equal to the arc length, rθ. This is why the same formula that measures a slice of a pie also tells you how far a bicycle travels per pedal stroke or how far the tip of a clock hand sweeps in an hour. If you know the rate of rotation — the angular velocity — multiplying by the radius converts it directly into linear speed along the arc. This deep connection between angle and distance is one of the reasons radians, and the tidy formula L = rθ, are so central to physics and engineering.
This calculator handles the arc length of a circular arc, which is by far the most common case and the one that has a simple closed formula, L = rθ. It is worth knowing, though, that "arc length" is a more general idea in mathematics: any curve — a parabola, an ellipse, the graph of a function — has an arc length too. For those curves there is no single tidy multiplication; instead, the length is found using integral calculus, by summing up countless tiny straight segments along the curve. The circular case is special because every point on a circle is the same distance from the center, which collapses that integral into the clean rθ relationship. So while this tool is built for circles and circular sectors — pie slices, curved tracks, gears, arches, fan shapes — the same underlying concept of "distance measured along a curve" extends to every shape in geometry. For circular arcs, the formula here gives an exact answer with no approximation needed.
Because an arc is just a fraction of the full circumference, a few reference angles make quick estimates easy. For a circle of radius r (circumference 2πr):
These fractions are a fast sanity check: if your central angle is a quarter of a full turn, your arc length should be about 1.57 times the radius. If the calculator gives you something wildly different, it is a sign that an angle unit or a value has been entered incorrectly.
The calculator's bidirectional design shines when you already know an arc length and need to work backwards. Suppose a curved garden border is 10 meters long and sweeps through a 120° angle around a central point, and you want the radius of that curve. Convert the angle: 120° = 2π/3 ≈ 2.0944 radians. Then rearrange L = rθ to r = L / θ = 10 / 2.0944 ≈ 4.77 meters. Enter the arc length and the angle into the calculator and it returns this radius instantly, along with the sector area and chord. This kind of reverse calculation is exactly what landscapers, fabricators, and designers need when the finished curved length is known but the underlying circle is not — and doing it by hand means remembering to convert to radians first, which the calculator handles for you.
Multiply the radius by the central angle in radians: L = r × θ. If the angle is in degrees, use L = 2πr × (θ ÷ 360). For a 15 cm radius and a 45° angle, the arc length is about 11.78 cm.
L = 2 × π × r × (θ ÷ 360), where θ is the central angle in degrees. This is the fraction of the full circumference (2πr) that the arc covers.
Yes. This calculator is bidirectional: enter the arc length together with either the radius or the angle, and it solves for the missing value using r = L / θ or θ = L / r (with θ in radians).
Arc length is the curved distance along the circle's edge between two points; the chord is the straight-line distance between those same two points. The chord (c = 2r × sin(θ/2)) is always shorter than the arc.
Both depend on the radius and the central angle. The sector area is A = r²θ/2 (θ in radians), and it can also be written as A = ½ × r × L, so once you know the arc length and radius you can find the sector area directly.
A radian is the angle for which the arc length equals the radius. A full circle is 2π radians (about 6.283), so one radian is about 57.30°. Radians make the arc length formula simply L = rθ.
This Arc Length Calculator is provided for educational and general informational purposes. It uses the standard circle geometry formulas; verify results independently for critical engineering, construction, or academic work.