Calculator Features
📐Six practical solve modes
🧮Step-by-step working
📏Degrees and radians
🌗Minor/major arc comparison
🚫Detects impossible input combinations
🔄Length & area unit conversion
🖨Printable / PDF results
📊CSV export
🖼SVG diagram export
📋Copy results & formula
🎚Decimal precision control
📱Mobile friendly
🧭 Jump to a section
What Is Arc Length?
Arc length is the distance measured along the curve of a circle, between two points on its edge — as opposed to the straight-line distance between the same two points. It's the actual "curved travel distance" you'd cover walking along that part of the circle, which is exactly why it matters for anything that follows a curved path: a running track, a road bend, a pipe fitting, a robot arm's toolpath.
💡 Did You Know? The radian, the natural unit for measuring angle in mathematics, is defined using arc length: one radian is the angle at which the arc length exactly equals the radius. That's why L = rθ has no extra scaling factor in radians, while the degrees version needs the θ/360 fraction to do the same job.
Arc vs Chord
These are easy to confuse but measure different things. The arc follows the curve; the chord is the straight line connecting the same two endpoints. Since a straight line is always the shortest path between two points, the chord is always shorter than the arc — the two only become equal in the limit as the angle shrinks toward zero.
Radius and Diameter
The radius (r) is the distance from the circle's centre to its edge; the diameter is twice that (d = 2r). Either can be used as a starting point for this calculator — if you only know the diameter, this calculator converts it to a radius internally before applying the arc length formula, so the result is identical either way.
Central Angle
The central angle is the angle at the circle's centre between the two radii marking the start and end of the arc. It's the single most important number for arc length calculations, since the arc is nothing more than "whatever fraction of the full circumference that angle represents."
Degrees vs Radians
Degrees divide a full circle into 360 equal parts — a convention with ancient Babylonian roots, not a mathematical necessity. Radians measure angle by arc length directly: a full circle is 2π radians (about 6.2832), and this direct relationship is what makes L = rθ so clean in radians. Engineering and physics calculations often default to radians for exactly this reason, while everyday and trade contexts (surveying, construction) usually stick with degrees.
Arc Length Formula
L = rθ (radians) | L = (θ/360) × 2πr (degrees)
Both express the same idea: the arc is whatever fraction of the full circumference (2πr) the central angle represents. A circle with radius 10 and a 60° arc covers 60/360 = 1/6 of the full circumference, giving an arc length of (1/6)×2π×10 ≈ 10.472 units.
Solving for Radius
r = L/θ (θ in radians)
Rearranging the arc length formula gives the radius directly from a known arc length and angle — useful when you've measured a curved distance on site (a road bend, a pipe segment) and need to work out what radius produced it.
Solving for Angle
θ = L/r (gives radians; multiply by 180/π for degrees)
If you know the radius and have measured the arc length, this gives the central angle directly — for example, working out how many degrees of a circular running track a particular straight-line marker corresponds to.
Sector Relationship
An arc is the curved boundary of a sector — the full "pie slice" bounded by the arc and its two radii. This calculator reports sector area alongside arc length since they share the same radius and angle, but if you need the sector's full perimeter, chord-versus-arc comparison, or a major/minor sector breakdown with both areas shown side by side, the dedicated Sector Calculator covers that in more depth.
Worked Examples
Running track — r=36.5 m, half-turn bend (θ=180°)
L = π×36.5 ≈ 114.668 m per bend — a standard 400 m track has two of these
Road curve — r=200 m, θ=30°
L = (30/360)×2π×200 ≈ 104.720 m of curved roadway
Roundabout — r=10 m, θ=90°
L ≈ 15.708 m of kerb line per quarter-turn
Pipeline bend — r=50 m, θ=45°
L ≈ 39.270 m of pipe needed for the bend section
Bridge curve — r=800 m, θ=10°
L ≈ 139.626 m — a gentle curve typical of a long-span bridge deck
CNC machining — r=8 mm, θ=120°
L ≈ 16.755 mm of toolpath for a rounded corner cut
Solving for radius — L=9.82, θ=22.5°
r = L/θ = 9.82/0.3927 ≈ 25 m — the spacing radius for 16 evenly spaced Ferris wheel gondolas
Chord + radius — the minor/major ambiguity — r=10, chord=10
Minor arc: θ = 2×arcsin(10/20) = 60°, giving arc length ≈ 10.472 m
Major arc: θ = 360°−60° = 300°, giving arc length ≈ 52.360 m — the same chord and radius, but a very different arc
This calculator asks which one you mean rather than silently picking one
Impossible-input example
Radius 10, arc length entered as 100: an arc can never be longer than its own circle's full circumference (≈62.832 for r=10)
The calculator correctly rejects this rather than returning a meaningless angle
Real-World Applications
| Field | Use |
|---|---|
| Civil engineering | Road curve design, horizontal alignment, curved kerb lengths |
| Construction | Curved formwork, arched openings, circular driveway aprons |
| Architecture | Curved facades, domes, amphitheatre seating arcs |
| Surveying | Boundary arcs, curved property lines, road reserve curves |
| Mechanical engineering | Belt and chain wrap length, cam and gear profiles |
| Manufacturing | CNC toolpath arcs, sheet-metal bend allowances |
| Road design | Highway and roundabout curve geometry |
| CAD design | Precise arc entities defined by radius and sweep angle |
| Robotics | Circular motion planning and arc-based toolpaths |
| Education | Teaching radians, circle geometry and proportional reasoning |
Glossary
- Arc
- A curved portion of a circle's circumference.
- Arc length (L)
- The distance measured along the arc, following its curve.
- Chord
- The straight line connecting the arc's two endpoints.
- Radius (r)
- The distance from the circle's centre to its edge.
- Central angle (θ)
- The angle at the centre between the two radii bounding the arc.
- Radian
- The angle at which arc length equals radius; a full circle is 2π radians.
- Minor arc
- The shorter of two arcs between two points on a circle, with a central angle strictly greater than 0° and less than 180°.
- Semicircular arc
- An arc with a central angle of exactly 180° — at this exact angle, both arcs between the two points are equal in length, so it's classified separately rather than as "minor."
- Major arc
- The longer of two arcs between the same two points, with a central angle strictly greater than 180° and less than 360°.
- Full-circle arc
- An arc with a central angle of exactly 360°, equal to the entire circumference. It has no separate complementary arc.
Common Mistakes
| Mistake | Fix |
|---|---|
| Using the chord length instead of the arc length | The chord is a straight line; the arc follows the curve and is always the longer of the two |
| Mixing degrees and radians | Convert consistently: degrees × π/180 = radians, radians × 180/π = degrees |
| Using the diameter where the radius is needed | Halve the diameter first, or use this calculator's Diameter + Angle mode directly |
| Forgetting there are two arcs for any two points | The "other" arc (360° − θ) is the major arc, unless θ is exactly 180° |
| Assuming arc length scales linearly with angle in degrees without the 360 fraction | Always divide by 360 (or use radians directly) — arc length is proportional to angle, not equal to it |
Formula Cheat Sheet
Quick Reference
Arc length:L = rθ (rad) = (θ/360)×2πr (deg)Radius:
r = L/θ | Central angle: θ = L/rChord:
c = 2r×sin(θ/2) | Sector area: A = ½r²θFull circle = 360° = 2π rad | Minor + major arc always sum to the full circumference
Practice Questions
Beginner (with answers)
- Find the arc length of a circle with radius 10 and central angle 90°.
- Find the arc length of a circle with radius 6 and central angle 60°.
- A circle has diameter 20. What is its radius?
- Convert 90° to radians.
- What fraction of a full circle is a 45° arc?
Show answers
1) (90/360)×2π×10≈15.708 2) (60/360)×2π×6≈6.283 3) r=20/2=10 4) 90×π/180=π/2≈1.5708 rad 5) 45/360=1/8
Advanced (with answers)
- A circle has radius 8 and an arc length of 10. Find the central angle in degrees.
- An arc has length 12 and central angle 40°. Find the radius.
- Find the chord length for an arc with radius 10 and central angle 120°.
- A circle has circumference 62.8 and central angle 60°. Find the arc length.
- Find the sector area for radius 12, central angle 90°.
Show answers
1) θ=10/8=1.25 rad≈71.62° 2) r=12/(40×π/180)≈17.19 3) c=2×10×sin(60°)≈17.321 4) r=62.8/2π≈9.994, L=(60/360)×62.8≈10.467 5) A=½×144×(π/2)≈113.097
🔑 Key Takeaways
- Arc length L = rθ (radians) or (θ/360)×2πr (degrees) — the curved distance along part of a circle.
- A chord (straight line) is always shorter than the arc between the same two points.
- The radian is defined so that arc length equals radius times angle directly, with no extra scaling.
- An arc is fully determined by a radius and central angle; diameter, circumference, chord or the arc length itself work equally well as alternative starting points.
- An arc is the curved edge of a sector — for the full pie-slice picture (perimeter, both sector areas), see the Sector Calculator.
Frequently Asked Questions
The distance measured along the curve of a circle between two points on its edge — the actual curved travel distance, not the straight-line distance between the same two points.
L = rθ in radians, or L = (θ/360)×2πr in degrees. A circle with radius 10 and a 60° arc has an arc length of about 10.472 units.
L = rθ (θ in radians), or equivalently (θ/360)×2πr (θ in degrees) — the fraction of the full circumference that the central angle represents.
L = (θ/360)×2πr — multiply the circumference by the fraction of 360° that the angle represents.
L = rθ — simply multiply the radius by the angle in radians, with no extra conversion factor needed.
r = L/θ, with θ in radians. If your angle is in degrees, convert it to radians first by multiplying by π/180.
θ = L/r, which gives the answer in radians. Multiply by 180/π to convert to degrees.
Arc length follows the curve of the circle; chord length is the straight-line distance between the same two points. The chord is always shorter than the arc, for any angle greater than 0°.
The shorter of two arcs between two points on a circle, with a central angle strictly between 0° and 180° — 180° itself is classified separately as a semicircular arc.
The longer of two arcs between the same two points, with a central angle strictly between 180° and 360°.
Yes — any two points on a circle divide it into exactly two arcs whose lengths always sum to the full circumference (2πr), and whose central angles always sum to 360°.
A chord and radius alone describe two different valid arcs — a shorter minor arc and a longer major arc — both with the same chord and radius but different arc lengths. This calculator asks you to choose rather than silently picking one, except when the chord equals the diameter, where both arcs are equal semicircles and the choice doesn't matter.
A full circle is 360° or 2π radians. Multiply degrees by π/180 to get radians, or radians by 180/π to get degrees.
One radian is defined as the angle at which the arc length exactly equals the radius — which is precisely why L = rθ works with no extra scaling factor in radians, unlike the degrees version.
Yes — this calculator's Diameter + Angle mode converts the diameter to a radius (r = d/2) automatically before applying the arc length formula.
Yes — this calculator's Circumference + Angle mode converts the circumference to a radius (r = C/2π) automatically before applying the arc length formula.
Yes — the central angle is found from θ = 2×arcsin(c/2r), and then the arc length follows from L = rθ. This gives the minor angle. For the major arc, use 2π − θ radians or 360° − θ degrees.
The full circumference (2πr), which occurs when the central angle is exactly 360° — at that point the "arc" is the entire circle.
(θ/360)×100 for degrees, or (θ/2π)×100 for radians — the central angle's share of the full circle.
An arc with a central angle of exactly 180° — exactly half the full circumference. It's classified on its own rather than as a minor or major arc, since both arcs formed by a diameter are equal in length.
A sector is the full "pie slice" region bounded by an arc and the two radii connecting its endpoints to the centre. The arc is just the curved boundary of that sector — see the dedicated Sector Calculator for the full sector picture, including perimeter and both sector areas.
A = ½r²θ — this calculator reports it alongside arc length since both depend on the same radius and central angle.
Road curve design and horizontal alignment calculations use arc length to determine the length of curved sections of road, based on the design radius and deflection angle.
Curved formwork, arched openings and circular driveway aprons all need arc length to determine material quantities for the curved sections.
Curved property boundaries and road reserve curves are measured and recorded using arc length, radius and central angle together.
Belt and chain wrap length around a pulley or sprocket is calculated using arc length, based on the contact angle and pulley radius.
Arc entities in CAD software and CNC toolpaths are defined by a radius and sweep angle, with the arc length determining machining time and material removal along curved cuts.
Circular motion planning for robot arms and mobile robots uses arc length to calculate travel distance along curved paths.
Any linear unit for the radius, diameter, chord and arc length itself (mm, cm, m, ft); sector area uses the squared version of the same unit; the central angle is in degrees or radians with no length unit.
Yes — choose any of the six modes above (radius+angle, diameter+angle, circumference+angle, radius+arc, arc+angle, or chord+radius), enter the known values, and the calculator solves everything else instantly with the working shown.
All six modes on this calculator are solved with exact closed-form formulas, accurate to floating-point precision — no approximation is needed for any arc length calculation.
Yes — enter the known values for whichever mode matches, and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
Using the chord length instead of the true arc length; mixing degrees and radians in the same calculation; using the diameter where the radius is needed; and forgetting the θ/360 fraction when working in degrees.
The calculator uses the required values for the selected mode to solve the arc, and cross-checks any extra values you entered (arc length or sector area) against the result, flagging a warning if they don't match.
Yes — use Copy Results to copy everything to the clipboard, Copy Formula for just the formulas used, Export CSV for a spreadsheet-ready file, Download SVG Diagram for a vector image, or Print / Save as PDF for a printable worksheet.
An arc only has two independent measurements (radius and central angle), so any two compatible values determine it fully. Six different natural combinations are offered because different situations start with different known values — sometimes a radius, sometimes a diameter or circumference, sometimes just a chord.
The calculator rejects it — an arc length can never exceed the full circumference of its own circle, since the arc is only ever a portion of that circumference.
No — an arc needs a positive central angle greater than 0° and no more than 360°; this calculator validates that range and explains why if it's violated.
No — an angle beyond 360° would simply wrap back around the same circle, so this calculator caps valid input at 360° and explains the limit if exceeded.
There's no fixed minimum size — any positive radius and any central angle between 0° and 360° describe a valid arc. Specific modes add their own constraints, like a chord never exceeding the diameter.
Yes — "arc length" and "curved distance" describe exactly the same measurement: the distance travelled along the curve rather than in a straight line.
The magnitude doesn't — travelling clockwise or counterclockwise between the same two points along the same arc gives the same length. Direction matters only for distinguishing the minor arc from the major arc between two points.
Radians are defined directly in terms of arc length (one radian is the angle giving an arc equal to the radius), which is why radian-based formulas in calculus, physics and engineering are consistently simpler than their degree-based equivalents.
Yes — set the central angle to 360°, and the arc length result will equal the full circumference, 2πr.
Arc length gives the true curved distance a vehicle or runner travels; chord length would understate that distance since it cuts straight across, which is why curved-path design always uses arc length for distance calculations.
Either works equally well — this calculator's Diameter + Angle mode simply halves the diameter to get the radius first. Use whichever measurement you actually have on hand; the result is identical either way.
Calculation Assumptions
ℹ️ What this calculator assumes
- Results are calculated using Euclidean (flat-plane) geometry.
- Measurements are assumed to lie in a flat plane, not on a curved or sloped surface.
- Rounding to the selected decimal precision may cause very small differences between displayed values and hand calculations.
- All input values are assumed to use the same unit — mixing units (e.g. a radius in metres and a chord in centimetres) will produce an incorrect result.
- All six solve modes have exact closed-form solutions — no numerical approximation is used anywhere on this page.
About This Calculator
Printable Formula Sheet
A one-page reference with every arc length formula on this page.
Arc Length Formula Sheet
MegaCalcOnline.com · Arc length, radius, angle and chord formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Arc length (radians) | L = rθ |
| Arc length (degrees) | L = (θ/360)×2πr |
| Radius from arc + angle | r = L/θ |
| Angle from arc + radius | θ = L/r |
| Chord length | c = 2r×sin(θ/2) |
| Sector area | A = ½r²θ |
| Radius from diameter | r = d/2 |
| Radius from circumference | r = C/(2π) |
References
- Khan Academy — Arc length and radians
- OpenStax Mathematics — circular measurement
- Maths Is Fun — Circle Arc
- NIST — SI units for length and area
- Wolfram MathWorld — Arc Length
Every formula and worked example on this page was independently verified — see “About This Calculator” above for the full review notes.