How this instrument works
Arc length answers a specific question: how long is the curved edge that a central angle sweeps out on a circle of a given radius? The formula is disarmingly short, s = rθ, but unlike most geometric identities this one isn't derived from something deeper — it is closer to a definition. The radian itself is defined so that an angle of exactly 1 rad cuts an arc exactly as long as the radius. Multiply that unit angle by r and you get r back; multiply a larger angle by r and the arc scales in step. There's no hidden derivation, because the relationship is baked into what 'radian' means in the first place.
The proportionality follows from a circle's own symmetry: rotating a fixed radius through equal angles always traces equal arcs, since every part of a circle looks like every other part after a rotation. Sweep a full turn, θ = 2π radians, and s becomes 2πr — the ordinary circumference, recovered as the special case of one complete revolution. Sweep half that angle and the arc is exactly half as long; the ratio never bends, because a circle has no straight stretch to complicate it.
The formula only works once θ is in radians, and that's the single most common place this trips people up by hand: plugging in 90 (degrees) instead of π⁄2 (radians) overstates the arc by a factor of roughly 57.3, since one radian is about 57.2958°. At the other edge, θ = 0 collapses the arc to a single point — zero length no matter how large the radius is — and a radius of 0 collapses every angle's arc to nothing as well, since there is no curve left standing to sweep.
- Enter the circle's radius into the Radius field, using any length unit — the arc length returns in that same unit.
- Type the sweep into Central angle, then pick its unit from the toggle; degrees, radians, and turns are all accepted and converted automatically.
- Read Arc length for the result: the curved distance that angle traces at that radius.
- To sanity-check a result, set Central angle to 360° (equivalently 2π rad or 1 turn) and confirm Arc length matches the plain circumference, 2πr.
Worked example — a quarter turn on a radius-2 circle
Set Radius to 2 and Central angle to 90°. Before multiplying, the sheet converts 90° to radians: π⁄2, or 1.5707963267948966. Then s = r × θ = 2 × 1.5707963267948966 = 3.141592653589793, and Arc length reports that figure — which is simply π.
Check it against the whole circle: a radius-2 circle has circumference 2π × 2 = 4π, about 12.566371. A 90° sweep is one quarter of a full 360° turn, so the arc should be one quarter of that circumference — 4π ⁄ 4 = π, matching the 3.141592653589793 the formula returned exactly. Both routes agree to the last digit, because both trace back to the same identity at θ = 2π.
Questions
Why does the arc length formula require the angle in radians?
Because the radian is defined that way: one radian is the angle whose arc equals the radius, so s = rθ needs no extra conversion constant only when θ is already in radians. Degrees carry a hidden factor — multiply degrees by π⁄180 first, or let this sheet's unit toggle handle it, or the arc will read roughly 57.3 times too long.
How is arc length different from a central angle calculation?
They are inverses of the same relationship, s = rθ. This sheet holds radius and angle fixed and solves for the arc; a central-angle calculator instead holds radius and arc fixed and solves for the angle, θ = s ⁄ r. Same identity, a different unknown isolated.
What happens to arc length at a full 360° turn?
It becomes the circumference. Setting the angle to 360° (2π radians, or 1 turn) reduces s = rθ to s = 2πr, the ordinary circumference formula — arc length is the general case, with the whole circle sitting inside it as one specific angle.
Does the formula still work for angles larger than 360°?
Yes — s = rθ has no built-in ceiling on θ. An angle of 720° (4π radians) describes two full laps and returns twice the circumference, which is exactly right for a rope wound twice around a spool or a wheel that has turned two complete revolutions.
What's the most common mistake people make finding arc length by hand?
Leaving the angle in degrees. A calculator multiplying s = rθ has no way of knowing whether the number entered is 90 (degrees) or 1.5708 (radians) — it simply multiplies. Typing 90 straight in gives an arc about 57.3 times too long; converting to radians first, or letting a unit-aware sheet do it, fixes the error.
Can an arc be longer than the circle's own diameter?
Easily. Once θ exceeds about 2 radians (roughly 114.6°), s = rθ already outgrows the diameter, 2r, and it keeps growing as θ increases while the diameter stays fixed. Only fairly small central angles produce an arc shorter than that straight-line width.