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Arc length calculator

Put a 60° angle and a radius of 10 into the arc length calculator and the arc is 10π/3, about 10.472. That is a sixth of the circumference, because 60° is a sixth of a full turn.

For the one-measurement mode.

A whole number gives answers exact in π, such as 36π.

For arcs and sectors, and for the equation of a circle. A root such as sqrt(5) or 3√2 stays exact.

In the unit below. A multiple of pi, such as 2pi/3, is read as radians and kept exact.

Which result leads. Every one is still calculated.

For the equation of a circle.

For center and radius from an equation. General form, or (x - 2)^2 + (y + 3)^2 = 25. Type squares as x^2 or x².

For the circle through three points.

Arc length10π/3 ≈ 10.471976

Central angle The angle between the two radii, at the center.
60° = π/3
Arc length s = rθ = 10 × π/3 = 10π/3 ≈ 10.471976. In degrees, s = (θ / 360) × 2πr = (60 / 360) × 2π × 10. θ must be in radians in s = rθ.
10π/3 ≈ 10.471976
Sector area A = ½r²θ = ½ × 10² × π/3 = 50π/3 ≈ 52.359878. In degrees, (θ / 360) × πr². Equivalently ½ × r × arc length.
50π/3 ≈ 52.359878
Chord length c = 2r sin(θ/2) = 2 × 10 × sin(30°) = 10. The straight line between the arc's ends, always shorter than the arc.
10
Segment area Segment = sector - triangle = ½r²θ - ½r² sin θ = 50π/3 - (25√3) = 50π/3 - 25√3 ≈ 9.0586074. The triangle is the one made by the two radii and the chord; past 180° its term is negative and the segment is larger than the sector.
50π/3 - 25√3 ≈ 9.0586074
Sagitta (arc height) h = r(1 - cos(θ/2)) = 10(1 - cos 30°) = 10 - 5√3 ≈ 1.339746. The height of the arc above the middle of the chord; with the center's distance to the chord it makes up the radius.
10 - 5√3 ≈ 1.339746
Inscribed angle Any angle with its vertex on the rest of the circle and its sides through the arc's ends is half the central angle (the inscribed angle theorem).
30° = π/6
Sector perimeter 2r + s: the two radii plus the arc.
20 + 10π/3 ≈ 30.471976
Every common angle at radius 10
Central angleArc lengthSector areaChordSegment area
30° (π/6)5.235987826.1799395.17638091.1799388
45° (π/4)7.853981639.2699087.65366863.9145691
60° (π/3)10.47197652.359878109.0586074
90° (π/2)15.70796378.53981614.14213628.539816
120° (2π/3)20.943951104.7197617.32050861.418485
180° (π)31.415927157.0796320157.07963
270° (3π/2)47.12389235.6194514.142136285.61945
360° (2π)62.831853314.159270314.15927
  • Exact forms appear where they exist: arc and sector are exact whenever the angle is a fraction of a turn, and the chord, segment and sagitta when half the angle is a multiple of 30° or 45°. Elsewhere the decimal is the answer.
  • Angles typed as a multiple of pi, such as 2pi/3, are read as radians whatever the unit says.

How the Circle Calculator works

The arc length formula is s = rθ, with the central angle θ in radians. With the angle in degrees it is s = (θ / 360) × 2πr. Both say the same thing: an arc is the same fraction of the circumference that its central angle is of a full turn. For radius 10 and 60°, s = (60 / 360) × 2π × 10 = 10π/3, about 10.472. In radians, 60° is π/3, and s = 10 × π/3, the same answer.

The radian form is short because of what a radian is. One radian is the angle whose arc is exactly one radius long, so an angle of θ radians cuts off θ radii of arc, and s = rθ is that definition restated. It is also the source of the commonest mistake: putting degrees into s = rθ. 10 × 60 = 600 cannot be an arc of a radius-10 circle, whose whole circumference is under 63. If the angle is in degrees, either convert it first (multiply by π/180) or use the degree form.

Running it backwards: θ = s / r gives the angle in radians from an arc length, and r = s / θ gives the radius. Choose "arc length" as the angle unit above, enter the arc and the radius, and the page finds the angle and everything else from it. An arc longer than the whole circumference, 2πr, is refused: no central angle goes past a full turn.

Arc length is not chord length. The chord is the straight line between the arc's ends, 2r sin(θ/2), and it is always shorter: 10 against 10.472 in the example. For small angles the two are nearly equal, which is why a short arc can be treated as straight. The gap grows to a factor of π/2, about 1.571, at a semicircle, where the arc is πr and the chord is the diameter, 2r.

Two points on a circle split it into two arcs. The minor arc has a central angle under 180° and the major arc has the rest, and their lengths add up to the circumference. For the 60° example, the major arc is 300°, which is 50π/3, about 52.36.

One caution about the search term itself. In a calculus course, "arc length formula" means the length of a curve y = f(x) between two points: the integral of √(1 + f′(x)²) dx. That is a different problem and this calculator does not do it. For a circle, that integral comes out to s = rθ, which is what is computed here.

Circle Calculator FAQ

What is the formula for arc length?

s = rθ, with θ in radians. A 60° angle is π/3 radians, so in a circle of radius 10 the arc is 10π/3, about 10.472.

How do I find arc length with the angle in degrees?

Use s = (θ / 360) × 2πr: the fraction of a full turn times the circumference. For 60° and radius 10 that is (1/6) × 20π = 10π/3. Or convert the angle to radians first by multiplying by π/180.

How do I find the central angle from the arc length?

Divide the arc by the radius: θ = s / r, in radians. Multiply by 180/π for degrees. An arc of 5 on a circle of radius 10 is 0.5 radians, about 28.648°.

What is the difference between arc length and chord length?

The arc follows the circle; the chord is the straight line between the same two points, 2r sin(θ/2). The chord is always shorter. For a 60° angle the chord equals the radius exactly, because the triangle it makes with the two radii is equilateral.

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