How this instrument works
Arc length measures the piece of a circle's boundary swept out by a specific central angle, using the formula arc length = r × θ, where θ must be expressed in radians rather than degrees. A radian is defined so that an angle of exactly one radian, at the center, cuts off an arc exactly one radius long — that built-in relationship is precisely why the multiplication works without any extra scaling constant. Enter degrees here and the sheet quietly converts them first; feed in radians directly and the conversion step simply vanishes.
This is a genuinely different question from the one a circumference sheet answers. Circumference reports the distance around the entire loop, the special case reached only when the swept angle completes a full 360°, or 2π radians. Arc length instead asks about a slice — a 90° turn, a 45° wedge, any portion smaller than one complete revolution. Set the angle to a full turn here and the two calculations agree exactly, but for anything short of that, arc length reports a smaller figure than the full boundary.
The relationship scales in a straightforward way: because arc length is a plain product of radius and angle, doubling either input doubles the result, and the swept distance grows in direct proportion to the fraction of a full revolution being traced. A hinge that opens 45°, an eighth of a turn, traces a path exactly one eighth as long as the full loop of the circle it belongs to — handy for anything from conveyor-belt wrap to laying out a curved garden bed.
- Enter the circle's radius into the Radius field.
- Enter the angle being swept into the Central angle field, choosing degrees, radians, or turns from the unit selector.
- Read Arc length for the exact distance traced along the rim by that angle.
- Set Central angle to a full 360° (or 2π radians) to check this figure against the circle's full circumference.
Worked example — a 90° slice of a radius-10 circle
A circle has a radius of 10 units, and the angle in question is a quarter turn: 90°, which is π⁄2 radians, or 1.5707963267948966 in full precision. Arc length is r × θ = 10 × 1.5707963267948966 ≈ 15.707963267948966 — call it 15.71 units for anything short of engineering tolerances. That is markedly shorter than the circle's full circumference of 2π × 10 ≈ 62.83, because a quarter turn sweeps only a quarter of the full rim: 15.71 sits almost exactly one fourth of 62.83.
Push the same radius to a full 360° sweep — θ = 2π ≈ 6.283185307179586 radians — and the swept distance becomes 10 × 6.283185307179586 ≈ 62.83, matching the circumference precisely, since a complete revolution is the one case where the two measures describe the same distance.
Questions
What is the formula for the arc length of a circle?
Arc length equals radius times the central angle in radians: s = rθ. With a radius of 10 and an angle of π⁄2 (a 90° turn), that gives 10 × 1.5707963267948966 ≈ 15.71. Degrees need converting first — multiply by π⁄180 — because the formula only balances when θ is a pure radian measure, itself defined as a swept distance divided by radius.
How is arc length different from circumference?
Circumference is the special case reached when the central angle equals a full turn — 360°, or 2π radians. A shorter sweep covers any slice smaller than that: a semicircle's path is half the circumference, a quarter-turn's path is a quarter, and so on. This sheet handles any angle; a plain circumference calculator only ever handles the full 2π case.
Why does the formula require radians instead of degrees?
Because a radian is itself defined by this relationship: one radian is the angle that sweeps out a path exactly one radius long. Multiplying radius by a degree figure would not carry that built-in ratio, so degrees must first be converted by the factor π⁄180. Enter degrees here and the conversion happens automatically before the multiplication runs.
Does the result depend on where the slice sits on the circle?
No — only the radius and the size of the central angle matter, not the slice's position or orientation. Rotate the same 90° wedge anywhere around the rim and the traced path is identical every time, since the formula never references a starting point, only the angle swept from the center.
Can the central angle be larger than 360°?
Yes, for a path that winds around more than once — think of a rope coiled twice around a drum. An angle of 720°, or 4π radians, simply produces a swept distance twice that of one full circumference, since the formula keeps working linearly past a single revolution with no upper cutoff.