Calculate the arc length and sector area of a circle from the radius and central angle, or solve for the radius or the angle instead.
Inputs
Choose what to solve for above — the input that becomes the unknown is hidden automatically.
Arc Length
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Enter a radius and angle above to compute.
Arc length—
Sector area—
Full circumference (2πr)—
Arc − chord gap—
Arc X, sector area Y.
Sector diagram
Solved Values
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Enter the values above to compute.
Radius—
Central angle—
Arc length—
Solve for radius or angle using the selector above.
Chord Length
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Enter the values above to compute.
Chord length—
Radius used—
Angle used—
The straight-line distance between the arc's two endpoints.
Related Calculators
4 min read3 steps6 terms3 examples6 FAQss = rθ
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Walk-through
How to Use This Calculator
3 steps▸
1
Enter the radius and central angle
Type the circle's radius and the central angle that sweeps out the arc you're measuring. Pick Degrees or Radians from the unit selector next to the angle field — the calculator converts internally so you never have to do that math by hand.
2
Read the arc length and sector area
The Arc & Sector tab updates instantly: the headline number is the arc length (s = rθ), and the detail line shows the sector area (½r²θ) and chord length together. Switch to the Chord tab for a dedicated readout of the straight-line distance between the arc's two endpoints.
3
Or solve backward for radius or angle
If you already know the arc length and want the radius or the angle instead, use the "Solve for" dropdown at the top. Choosing "Radius" swaps the radius field for an arc-length field; choosing "Angle" does the same for the angle field. The Solve tab then shows whichever value you're solving for.
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Reference
Formula & Methodology
3 formulas▸
Arc length
s = rθ
Arc length s equals the radius r multiplied by the central angle θ, measured in radians. This is the definition of a radian: one radian is the angle that sweeps out an arc exactly as long as the radius, so multiplying by r simply scales that relationship up. If your angle is in degrees, convert it first with θ(rad) = θ(deg) × π/180.
Sector area
A = ½r²θ
A circular sector is the pie-slice-shaped region bounded by the two radii and the arc. Its area is half the radius squared times the central angle in radians — the same ½r²θ shape as the full-circle area formula πr² (which is just the θ = 2π special case: ½r²(2π) = πr²).
Chord length
c = 2r·sin(θ/2)
The chord is the straight line connecting the arc's two endpoints, as opposed to the curved arc itself. It comes from splitting the isosceles triangle formed by the two radii and the chord into two right triangles: each half-chord is r·sin(θ/2), so the full chord is twice that.
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Glossary
Key Terms Explained
6 terms▸
Arc lengthThe distance along the curved part of a circle between two points, as opposed to the straight-line (chord) distance between the same two points. Computed as s = rθ with θ in radians.
SectorThe pie-slice-shaped region of a circle bounded by two radii and the arc between them. Its area is ½r²θ.
Central angleThe angle formed at the circle's center by the two radii that bound a sector or arc. Can be measured in degrees or radians — this calculator accepts and converts both.
RadianThe SI unit of angle, defined so that an angle of 1 radian sweeps out an arc equal in length to the radius. A full circle is 2π radians (about 6.283), which equals 360°, so 1 radian ≈ 57.296°.
ChordA straight line segment connecting two points on a circle. Unlike the arc, which follows the curve, the chord is the shortest (straight-line) path between the two points: c = 2r·sin(θ/2).
Circular sectorAnother name for a sector — the region enclosed by an arc and the two radii at its ends, commonly visualized as a slice of pie or pizza.
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Scenarios
Real-World Examples
3 worked examples▸
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Forward calculation
Radius 5, central angle 60°
Radius 5Central angle 60°
Converting 60° to radians gives θ = 60 × π/180 ≈ 1.0472 rad. Arc length = 5 × 1.0472 ≈ 5.236 units. Sector area = ½ × 5² × 1.0472 ≈ 13.09 sq units. Chord length = 2 × 5 × sin(30°) = 10 × 0.5 = 5 units — notice the chord (5) is shorter than the arc (5.236), as it always is for any angle less than a full circle, since a straight line is the shortest path between two points.
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Solving backward for radius
Arc length 5.236, central angle 60° → radius?
Arc length 5.236Central angle 60°
Rearranging s = rθ gives r = s / θ. With θ = 60° ≈ 1.0472 rad, radius = 5.236 / 1.0472 ≈ 5.000. This is exactly the reverse of the first example — useful when you've measured an arc's length in the field (say, a curved section of road or track) and the angle it subtends, but need to find the radius of the curve it lies on.
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Chord from a wider sweep
Radius 10, central angle 90°
Radius 10Central angle 90°
At 90° (a quarter circle), θ = π/2 ≈ 1.5708 rad. Arc length = 10 × 1.5708 ≈ 15.71 units. Chord length = 2 × 10 × sin(45°) ≈ 20 × 0.7071 ≈ 14.14 units. Here the gap between arc and chord is much more visible than in the 60° example — the wider the angle, the more the curved arc outpaces the straight chord, until at a full 360° sweep the chord shrinks back to 0 while the arc equals the entire circumference.
An arc is just a piece of a circle's circumference, and once you know the radius and the central angle that defines it, three related measurements fall out immediately: how long the arc itself is, how much area the pie-slice-shaped sector covers, and how far apart the arc's two endpoints are in a straight line (the chord). This calculator computes all three from radius and angle, and can also work backward to solve for the radius or the angle when you already know the arc length.
How the Arc Length Calculator works
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The core relationship is s = rθ, where θ must be in radians — this is literally the definition of a radian, since one radian is the angle that traces an arc exactly as long as the radius. If you enter your angle in degrees, the calculator converts it to radians (θ_rad = θ_deg × π/180) before running any formula, then converts back for display so you never see an unfamiliar unit unless you ask for it.
Sector area (½r²θ) and chord length (2r·sin(θ/2)) both fall out of the same known radius and angle, so all three results update together the moment you change an input. Solving backward for radius (r = s/θ) or angle (θ = s/r) just rearranges the same s = rθ formula.
Inputs and what they mean
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Radius is the distance from the circle's center to its edge — any consistent unit works (inches, meters, miles) as long as you're consistent. Central angle is how wide the pie-slice is, in either degrees (the everyday unit, where a full circle is 360°) or radians (the mathematical unit, where a full circle is 2π ≈ 6.283). Arc length is the curved distance along the circle's edge that the angle sweeps out.
The "Solve for" selector controls which of these three is the unknown: leave it on "Arc & sector" to compute forward from radius and angle, or switch it to "Radius" or "Angle" to work backward from a known arc length.
Limits and edge cases
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Radius and angle must both be positive numbers — a zero or negative radius or angle doesn't describe a real arc. When solving for the angle from a known arc length and radius, the arc length can't exceed the circle's own circumference (2πr), since a single arc can't be longer than the whole circle it belongs to; the calculator flags this rather than returning an angle greater than 360°.
At exactly a full circle (θ = 2π radians = 360°), the chord formula still works but becomes trivial — the two "endpoints" of a full circle are the same point, so the chord length is 0, while the arc length equals the full circumference 2πr. This calculator handles plane (2D, Euclidean) circles only; it isn't meant for arcs on a sphere or other curved surface.
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Questions
Frequently Asked Questions
6 questions▸
What formula does this calculator use for arc length?+
Arc length s = rθ, where r is the radius and θ is the central angle in radians. If you enter degrees, the calculator converts to radians first (θ_rad = θ_deg × π/180) since the formula only works in radians.
What's the sector area formula?+
Sector area = ½r²θ, with θ again in radians. This is the same shape as the full-circle area formula πr² — plugging in θ = 2π (a full circle) gives ½r²(2π) = πr², confirming the formula is consistent at the boundary.
How do I convert degrees to radians?+
Multiply the degree value by π/180 (approximately 0.017453). For example, 60° × π/180 ≈ 1.0472 radians. This calculator does the conversion automatically — just pick Degrees or Radians from the unit selector next to the angle field.
What's the chord length formula, and how is it different from arc length?+
Chord length = 2r·sin(θ/2). The chord is the straight-line distance between the arc's two endpoints, while the arc length is the distance along the curve. The chord is always shorter than (or, at θ = 0, equal to) the arc, since a straight line is the shortest path between two points.
Can this calculator solve for the radius or the angle instead of the arc?+
Yes. Use the "Solve for" dropdown to switch the calculator into reverse mode: choose "Radius" to solve r = s/θ from a known arc length and angle, or choose "Angle" to solve θ = s/r from a known arc length and radius.
What happens at a full 360° circle?+
At θ = 2π radians (360°), the arc length formula gives s = 2πr — the familiar circumference formula — and the sector area gives A = πr², the familiar circle-area formula. The chord length drops to 0, since the arc's "two endpoints" are actually the same point on a full circle.
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