Arc Length Calculator
Calculate, understand, and apply arc length formulas accurately for circles, geometry, trigonometry, engineering, construction, design, and educational problem-solving. Enter known values in the tool below, then read the formulas and examples on this page.
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Arc Length Calculator
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Result
Arc Length
m -
Enter at least two known values, then calculate.
What Is Arc Length?
Arc length is the curved distance measured along the boundary of a circle between two points on the circumference. When those two points and the center form a sector, the arc is the curved edge opposite the central angle.
Arc length is not chord length. The chord is the straight segment connecting the arc endpoints. The arc follows the circle, so it is longer than the chord for any nonzero central angle.
You will use arc length in geometry homework, mechanical design, road layout, pipe bending, and any task where a circular path must be measured precisely. The calculator above returns arc length when you supply any two related measurements.
- Definition: curved distance along a circular arc
- Arc of a circle: part of the circumference bounded by a central angle
- Real-world use: gears, bridges, tracks, ducts, and circular layouts
Arc Length Formula
When the central angle θ is given in degrees and the radius r is known, the standard arc length formula is:
L = (θ / 360°) × 2πr
This formula states that arc length is a fraction of the full circumference. The fraction equals the central angle divided by 360°. Multiply that fraction by 2πr to get the curved distance.
If you know diameter d instead of radius, replace r with d/2. If you know sector area A and radius, you can also use L = 2A/r because A = ½r²θ in radians.
L = (θ / 360°) × 2πr
Degree-based formula. θ is the central angle in degrees, r is the radius in your chosen linear unit.
L = (θ / 360°) × πd
Same relationship written with diameter d = 2r.
L = 2A / r
Find arc length from sector area A and radius when the angle is not given directly.
Arc Length Formula in Radians
In radian measure, the arc length formula simplifies to:
L = rθ
Here θ must be in radians, not degrees. One radian is the angle where the arc length equals the radius. This is the natural form used in trigonometry and calculus because no conversion factor is needed.
To convert degrees to radians, multiply by π/180. Example: 90° = π/2 radians. Then L = r × (π/2).
L = rθ (θ in radians)
Direct relationship between radius and subtended angle.
θ_rad = θ_deg × (π / 180)
Convert a degree measurement before applying L = rθ.
How to Calculate Arc Length
- Identify the radius r (or convert from diameter using r = d/2).
- Measure or read the central angle θ at the circle center. Confirm whether it is in degrees or radians.
- Choose the correct formula: L = (θ/360°) × 2πr for degrees, or L = rθ for radians.
- Substitute values and compute. Keep all linear units consistent (meters with meters, feet with feet).
- Verify the result: arc length must be less than or equal to the full circumference 2πr for a single sector up to 360°.
Arc Length Examples
Degree example: 60° at r = 6 m
L = (60/360) × 2π(6) = (1/6) × 12π ≈ 6.283 m
Radian example: θ = π/3 at r = 5 m
L = 5 × (π/3) ≈ 5.236 m (same as 60° because π/3 rad = 60°)
From diameter: 90° at d = 10 m
r = 5 m, L = (90/360) × 2π(5) = 0.25 × 10π ≈ 7.854 m
Engineering check: quarter circle, r = 2.5 m
θ = 90°, L = (90/360) × 2π(2.5) ≈ 3.927 m for duct or rail layout
Arc Length and Central Angle
The central angle is the angle formed at the circle center by the two radii that bound the sector. Arc length grows proportionally with this angle when radius is fixed.
Doubling the central angle doubles the arc length. At 360°, the arc becomes the full circumference. At 180°, it is a semicircle with L = πr.
- θ in degrees: use L = (θ/360°) × 2πr
- θ in radians: use L = rθ
- Convert with θ_rad = θ_deg × π/180 before mixing formulas
Arc Length vs Chord Length
Arc length follows the curve; chord length cuts straight across. For the same sector, arc length is always greater than chord length when the central angle is greater than zero.
Chord length c = 2r sin(θ/2) when θ is in radians. Our calculator accepts chord length as an input and returns arc length when paired with radius or angle.
Arc: L = rθ
Curved distance along the circumference.
Chord: c = 2r sin(θ/2)
Straight-line distance between arc endpoints.
Arc Length from Diameter
Drawings often label diameter instead of radius. Convert first: r = d/2. Then apply the standard formula.
Example: d = 8 m and θ = 45° gives r = 4 m and L = (45/360) × 2π(4) = 0.125 × 8π ≈ 3.142 m.
r = d / 2
Convert diameter to radius before calculating.
L = (θ / 360°) × πd
Arc length directly from diameter and angle in degrees.
Common Arc Length Calculation Mistakes
Using degrees in L = rθ
The formula L = rθ requires radians. Convert degrees first or use L = (θ/360°) × 2πr instead.
Mixing radius and diameter
Pick one. If both appear on a diagram, convert d to r = d/2 before substituting.
Confusing arc with chord
A straight measurement across the sector is chord length, not arc length.
Ignoring units
Keep radius, arc, and chord in the same unit. Area must use the square of that unit.
Arc Length and Sector Area
Sector area A and arc length L both depend on the same central angle and radius. Area A = ½r²θ (θ in radians). Combining with L = rθ gives a useful shortcut: L = 2A/r when you know area and radius but not the angle.
This link helps in design problems where shaded area is given on a plan but the curved boundary length is what you need to fabricate or estimate material.
A = ½ r² θ
Sector area with θ in radians.
L = 2A / r
Arc length from sector area and radius without finding θ first.
FAQs About Arc Length
- What is the main arc length formula?
- In degrees: L = (θ/360°) × 2πr. In radians: L = rθ. Both give the same curved distance when θ is converted correctly.
- How many values do I need to use the calculator?
- Enter any two known measurements among central angle, radius, diameter, sector area, and chord length. The tool selects the correct formula.
- Is arc length the same as circumference?
- Only when the central angle is 360°. For smaller angles, arc length is a portion of the full circumference 2πr.
- Can I calculate arc length from chord length alone?
- You need a second value such as radius or central angle. Chord and radius together determine the angle, then arc length follows.
- Where is arc length used in real projects?
- Road curves, conveyor paths, pipe bends, architectural arches, gear profiles, and survey layouts all rely on accurate arc distance.
- Does the calculator store my inputs?
- No. All math runs locally in your browser. Nothing is uploaded to a server.