Arc Length Calculator

Background.

The Quanta arc length calculator finds the length of a curved slice of a circle's edge — an arc — together with the area of the pie-slice-shaped sector it bounds, from just a radius and a central angle. Enter the angle in either degrees or radians and the calculator converts automatically, returning arc length, sector area, and the angle re-expressed in radians for reference.

Where Quanta's area-circle and circumference calculators answer questions about a whole circle, this calculator is built for the partial-circle case: whenever the shape you actually care about is a slice rather than the full disc. That distinction matters constantly in practice. A pizza cut into eight equal slices has each slice spanning a 45° central angle; the crust length of one slice is an arc length, and the amount of pizza in it is a sector area. A road engineer designing a curved section of highway specifies a radius and a deflection angle, and the arc length is exactly the length of pavement that curve will require. A plumber bending a length of conduit through a 90° elbow needs the arc length of the bend to know how much pipe the bend consumes. A landscaper laying out a curved garden bed along one edge of a circular lawn needs the arc length for edging material and the sector area for sod. Mechanical designers computing the tooth spacing on a gear, animators easing a camera along a circular path, and data-visualisation designers sizing a wedge of a pie chart are all, underneath the specific vocabulary of their field, solving the same two-line arc-length-and-sector-area problem.

The central angle can be entered in degrees, the everyday unit most people learn in school, or in radians, the unit calculus and physics use natively because it makes the arc length formula exactly L = rθ with no conversion constant at all. The calculator restricts the angle to at most one full turn — 360° or 2π radians — because a sector, by definition, cannot wrap around the circle more than once; if you need the length of a multi-turn spiral or helix, that is a different geometric object and a different formula. Below the calculator you will find the classical derivation connecting central angle to arc length (going back to Euclid's proof that equal central angles subtend equal arcs), a fully worked quarter-circle example, and answers to the questions people ask most often about finding arc length and sector area.

What is arc length calculator?

An arc is a connected portion of a circle's circumference — the curved path you get by walking partway around the circle rather than all the way. The central angle θ is the angle formed at the circle's centre by the two radii drawn to the arc's endpoints; it determines exactly how much of the circle the arc covers. When θ is measured in radians — the ratio of arc length to radius, by definition — the arc length formula is simply L = r × θ: multiply the radius by the angle. This is why radians are the natural unit for this formula; using degrees requires first converting with θ_radians = θ_degrees × π/180.

The sector is the pie-slice-shaped region enclosed by the arc and the two radii bounding it. Its area is proportional to the fraction of the full circle the central angle represents: sectorArea = (θ / 2π) × πr² = (1/2) × r² × θ, again with θ in radians. Both formulas reduce to the familiar full-circle formulas when θ = 2π radians (360°): arc length becomes 2πr, the circumference, and sector area becomes πr², the full circle's area — a useful sanity check built into the worked example below. Because a sector cannot represent more than a complete revolution of the circle, the central angle is restricted to the interval (0°, 360°] or, equivalently, (0, 2π] radians.

How to use this calculator.

1. 01Enter the radius of the circle. Zero is accepted (a degenerate point with no arc), but negative radii are rejected as physically meaningless.
2. 02Enter the central angle that the arc spans — the angle at the centre of the circle between the two radii drawn to the arc's endpoints.
3. 03Select whether that angle is in degrees or radians using the unit dropdown. The calculator converts internally, so you never need to do the conversion yourself.
4. 04Read the primary output, arc length, in the same linear unit as your radius (metres in, metres out; feet in, feet out).
5. 05Check the sector area output for the pie-slice area bounded by the arc and the two radii, reported in square units of your radius's unit.
6. 06The angle-in-radians output is shown for reference — useful if you need to feed the value into a formula, spreadsheet, or CAD tool that expects radians.
7. 07For a full circle's circumference or area rather than a partial arc or sector, use Quanta's circumference calculator or area of a circle calculator instead.

The formula.

The arc length formula follows directly from the definition of the radian. One radian is defined as the central angle that subtends an arc exactly as long as the circle's radius — so an angle of θ radians subtends an arc θ times as long as the radius, giving L = r × θ with no conversion constant needed. This proportionality between central angle and arc length is exactly what Euclid proves in Book III, Proposition 26 of the Elements: in equal circles, equal central angles stand on equal arcs. Modern radian measure simply makes that ancient proportionality numerically explicit — double the angle and you double the arc, at any radius, because the ratio arc-length-to-radius is what defines the angle in the first place.

If your angle is measured in degrees, convert first: since a full 360° turn corresponds to 2π radians, one degree equals π/180 radians, so θ_radians = θ_degrees × π/180. Substituting into L = rθ gives the degrees version directly: L = r × θ_degrees × π/180.

Sector area follows from a proportionality argument rather than a fresh derivation. A sector with central angle θ (radians) is the fraction θ/(2π) of the whole circle, because 2π radians is a complete turn. The whole circle's area is πr², so the sector's area is that fraction of the whole: sectorArea = (θ/2π) × πr². The π cancels, leaving sectorArea = (1/2) × r² × θ. Both formulas contain the same core quantity — θ in radians multiplied against powers of r — which is why arc length and sector area are always computed together: once θ_radians is known, both follow in one line each. Setting θ to a full turn (2π radians, 360°) recovers the whole-circle formulas exactly: arc length becomes 2πr (the circumference) and sector area becomes πr² (the full area), confirming the sector formulas are strict generalisations of the whole-circle ones rather than approximations.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book III, Proposition 26 — in equal circles, equal angles stand on equal arcs whether at the centre or the circumference, the classical basis for arc length being proportional to central angle (David E. Joyce edition, Clark University).
2. [2]NIST Digital Library of Mathematical Functions, §4.14 — definitions and periodicity of the circular (trigonometric) functions, including radian-measure conventions.
3. [3]Weisstein, Eric W. "Circular Sector." MathWorld — A Wolfram Web Resource. Gives the arc length formula s = Rθ and the sector area formula A = (1/2)R²θ used by this calculator.
4. [4]Coxeter, H. S. M. Introduction to Geometry, 2nd edition. Wiley, 1969 — classic reference textbook covering radian measure, arc length, and circular sectors.