Arc Length & Sector Area Calculator

Understanding the arc length and sector area calculator — concept and principle

An arc is a piece of a circle’s edge, and a sector is the pie-slice region between two radii and that arc. Given the radius and the central angle, this calculator returns the arc length, the sector area, the chord (the straight line joining the arc’s ends) and the segment area (the region between the chord and the arc). These numbers are used for curved paths and roads, bent metal strips, fan blades, pie charts, windows with arched tops and the swept area of wipers or doors.

Turn 60° into π/3 radians first; then s = rθ and sector = ½r²θ.

How it is calculated

First convert the angle to radians: θ = degrees × π / 180.

Arc length: s = rθ
Sector area: A = ½r²θ
Chord length: c = 2r sin(θ/2)
Segment area: ½r²(θ − sin θ)

The arc and chord have the unit of r; the two areas have that unit squared.

The principle

A radian is defined so that an angle of 1 rad cuts an arc equal to the radius. A full turn is 2π rad, matching the full circumference 2πr, so any arc is simply the same fraction of the circumference: s = rθ. The sector is the same fraction of the full area: (θ / 2π) × πr² = ½r²θ. The chord comes from splitting the isosceles triangle formed by the two radii into two right triangles, each with hypotenuse r and half-angle θ/2, so half the chord is r sin(θ/2). The segment is the sector minus that triangle, whose area is ½r² sin θ.

Worked example

Inputs: r = 10, angle = 60°, so θ = 60 × π / 180 = π/3 ≈ 1.047198 rad.
Arc length: 10 × 1.047198 = 10.47197551.
Sector area: ½ × 100 × 1.047198 = 52.35987756.
Chord: 2 × 10 × sin 30° = 20 × 0.5 = 10.
Triangle area: ½ × 100 × sin 60° ≈ 43.30127; segment = 52.35988 − 43.30127 = 9.058607371.

Each point is the calculator re-run with only “Radius” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Radius	Arc length
2.5	2.617993878
5	5.235987756
7.5	7.853981634
10	10.47197551
12.5	13.08996939
15	15.70796327
20	20.94395102

How to read the result

At 60° the chord equals the radius, because the triangle formed is equilateral. The arc is always a little longer than the chord; the gap is tiny for small angles and grows as the angle widens. At 180° the chord becomes the diameter and the segment is a half-disc. A small segment area relative to the sector means the arc is nearly flat.

Common mistakes

Putting degrees directly into s = rθ; the formula needs radians.
Mixing up the sector (the pie slice to the centre) with the segment (the part cut off by the chord).
Using the diameter instead of the radius.

Keep in mind: The results assume a true circular arc. Curves in roads, rails and bent parts are often transition curves or have material thickness, so real designs need the relevant standards and a qualified professional.

Frequently asked questions

How do I find the angle from the arc length?

θ = s ÷ r in radians, then multiply by 180/π for degrees. An arc of 5 on a radius of 10 is 0.5 rad, about 28.65°.

What is the arc length of a full circle?

For 360° (2π rad) the arc length is 2πr, the circumference.

Can the angle be more than 180°?

Yes. All formulas still work up to 360°; for a major segment (more than half the disc), θ − sin θ simply becomes larger.

Sources: BIPM, The International System of Units (SI Brochure), 9th edition, 2019 (definition of the radian) · Euclid, Elements, Book III (properties of chords and arcs)

Checked 8 Nov 2026. Standard formulas, computed in your browser with double-precision numbers.

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