An arc is just a piece of a circle's edge, and its length turns up in more places than geometry class: the stagger on a running track, the pitch of gear teeth, the curve of a road, the sweep of a windshield wiper. The arc length calculator works out how long that curved piece is from two things you usually already know — the radius of the circle and the angle swept out at its centre. Give it those two values and it returns the distance along the curve itself, not the straight-line shortcut across it.
Quick Answer
Arc length is s = rθ when the angle θ is measured in radians. If your angle is in degrees, use s = (θ ÷ 360) × 2πr instead. In both versions, r is the radius and the answer comes out in the same unit as the radius.
The Geometry Behind an Arc
The whole circumference of a circle is 2πr, and an arc is a fraction of that whole. A 60° arc is one sixth of the circle because 60 is one sixth of 360, so its length is one sixth of the circumference. That proportion is all the degrees formula is doing: it takes the fraction θ/360 of the full turn and multiplies it by the full circumference.
Radians make the same idea cleaner. A radian is defined so that an angle of 1 radian cuts an arc exactly as long as the radius. A full turn is 2π radians, so an angle of θ radians cuts an arc of θ radii — hence s = rθ, with no conversion factors at all. This is why mathematicians and engineers prefer radians for anything involving rotation.
Two consequences are worth remembering. First, arc length grows in direct proportion to both the radius and the angle: double either one and the arc doubles. Second, the arc is always longer than the chord — the straight segment joining its endpoints — because the arc bulges outward. Confusing those two distances is a frequent source of wrong answers in construction layouts.
How to Use the Arc Length Calculator
- Measure or look up the radius of the circle — the distance from the centre to the edge, which is half the diameter.
- Find the central angle, the angle at the centre between the two radii that bound your arc.
- Check which unit your angle is in, degrees or radians, and enter it in the matching form of the calculation.
- Enter the radius, keeping its unit in mind (centimetres, metres, inches).
- Read the arc length; it is expressed in that same unit.
- Sanity-check the result: it must be shorter than the full circumference, 2πr.
Worked Example: A 60° Arc on a Radius-10 Circle
Suppose a circular garden path has radius 10 metres, and you need the length of edging for a section spanning 60°. Using the degrees formula: s = (60 ÷ 360) × 2 × π × 10 = (1/6) × 62.83 = 10.47 metres.
Now check with radians. 60° converts to π/3 radians, which is about 1.0472. Then s = rθ = 10 × 1.0472 = 10.47 metres. The two routes agree, as they must — they are the same formula wearing different units. Notice the arc (10.47 m) is a little longer than the radius itself and far shorter than the full circumference of 62.83 m, which is the right neighbourhood for a sixth of the circle.
- Radius 8 cm, angle 90° → 12.57 cm
- Radius 12 m, angle 45° → 9.42 m
- Radius 5 cm, angle 2 radians → 10 cm exactly
- Radius 15 cm, angle 180° → 47.12 cm (half the circumference)
Radians or Degrees? Getting the Angle Right
Nearly every arc length error is a units error. If you feed 60 into s = rθ without converting, you are claiming the angle is 60 radians — more than nine full turns — and the answer comes out absurd. Either convert degrees to radians first (multiply by π/180) or use the degrees version of the formula. A second trap is using the diameter where the radius belongs, which doubles the answer. A third is measuring the angle at the wrong point: the formula needs the central angle, between two radii. An angle marked on the circumference of the circle is half the central angle for the same arc — a classic circle-theorem result that quietly halves or doubles answers when it is missed. Keep the radius in one consistent unit from start to finish as well, since the answer inherits that unit.
Where Arc Lengths Matter
Race officials use arc length to stagger starting positions on curved tracks, so every runner covers the same total distance despite the outer lanes being longer per lap. Belt and chain drives depend on the arc of contact between belt and pulley — the longer that wrapped arc, the more grip the drive has before it slips. Irrigation engineers calculate the arc a rotating sprinkler covers, and road designers work with arc lengths whenever a route bends, because vehicles travel along the curve, not across it.
A clock makes the idea concrete at home. The tip of a 15 cm minute hand sweeps through 120° in twenty minutes, a third of the dial. Its arc is (120 ÷ 360) × 2 × π × 15 = about 31.4 cm — the tip travels nearly a third of a metre in those twenty minutes even though the hand itself is only 15 cm long, because it describes the circle's edge rather than crossing it. In each of these cases the straight-line distance would understate the real quantity, which is exactly the gap this calculator closes.
Keep Going Around the Circle
An arc usually comes with a slice of the circle attached. The Area of a Sector Calculator finds the area of that slice from the same radius and angle, while the Circumference Calculator covers the complete circle when your arc turns out to be the whole thing. Both sit alongside this tool in the math section.