Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Geometry Calculators > Arc Length Calculator

Arc Length Calculator

Calculate arc length, radius, or central angle of a circle sector with interactive diagram, step-by-step solution, and support for degrees and radians.

Arc Length Calculator

Quick Examples:

r=5, θ=45° r=10, θ=90° r=3, θ=π/2 arc=7.854, θ=90° r=6, arc=9.42
r = 5 θ = 45° arc = ?
Arc Length
Radius
Angle
Tip: Use the interactive diagram above to visualize your sector. The diagram updates as you enter values.

Embed Arc Length Calculator Widget

About Arc Length Calculator

Welcome to the Arc Length Calculator, a comprehensive geometry tool that calculates arc length, radius, or central angle of a circle sector. With an interactive SVG diagram, step-by-step mathematical solutions, and support for both degrees and radians, this calculator helps students, engineers, and professionals solve arc-related problems quickly and accurately.

What is Arc Length?

Arc length is the distance along the curved path of an arc - a portion of a circle's circumference. It represents the actual length you would measure if you laid a string along the curved edge of a sector. Arc length is fundamental in geometry, trigonometry, and has numerous real-world applications in engineering, architecture, and physics.

The arc length depends on two factors: the radius of the circle and the central angle that the arc subtends. A larger radius or larger angle produces a longer arc.

Arc Length Formula

Arc Length (radians)
$$s = r \times \theta$$

When the angle is given in degrees:

Arc Length (degrees)
$$s = \frac{\pi \times r \times \theta}{180}$$

Where:

Related Formulas

From the arc length formula, we can derive formulas for finding radius and angle:

Radius from Arc Length
$$r = \frac{s}{\theta}$$ (angle in radians)
Angle from Arc Length
$$\theta = \frac{s}{r}$$ (result in radians)

How to Use This Calculator

  1. Select calculation mode: Choose what you want to find - Arc Length, Radius, or Angle.
  2. Enter known values: Input the values you know. The calculator will disable the field for the value being calculated.
  3. Select angle unit: Choose between degrees or radians for the angle input/output.
  4. Calculate: Click the Calculate button to see the result along with all related values.
  5. Review steps: Examine the step-by-step solution showing the mathematical process.

Understanding the Diagram

The interactive SVG diagram shows:

The diagram updates in real-time as you change the input values, providing visual feedback for your calculations.

Converting Between Degrees and Radians

Understanding the relationship between degrees and radians is essential for arc length calculations:

Common conversions:

Applications of Arc Length

Engineering and Construction

Arc length calculations are essential in designing curved roads, railway tracks, bridges, and architectural elements like domes and arches. Engineers use these formulas to determine material requirements and ensure proper curvature.

Manufacturing

In manufacturing, arc length is used for designing gears, pulleys, cam mechanisms, and any circular or curved components. It helps determine belt lengths, gear tooth profiles, and circular saw blade specifications.

Navigation

Arc length is used to calculate distances on Earth's surface (which is approximately spherical). Great circle distances between two points are essentially arc lengths on a sphere.

Physics

In physics, arc length is crucial for analyzing rotational motion, pendulum movement, and circular trajectories. It connects angular measurements to linear distances traveled.

Computer Graphics

Arc length parameterization is important in computer graphics for rendering smooth curves, animating objects along paths, and creating fonts and vector graphics.

Arc Length vs. Other Measurements

Arc Length vs. Chord Length

The chord is the straight line connecting the two endpoints of an arc. Arc length is always greater than chord length (except for a degenerate arc of 0 length). The chord formula is:

Chord Length
$$c = 2r \sin\left(\frac{\theta}{2}\right)$$

Arc Length vs. Sector Area

While arc length measures the curved boundary, sector area measures the enclosed region. The sector area formula is:

Sector Area
$$A = \frac{1}{2}r^2\theta$$ (angle in radians)

Frequently Asked Questions

What is the arc length formula?

The arc length formula is s = r × θ, where s is the arc length, r is the radius, and θ is the central angle in radians. When the angle is in degrees, use s = (π × r × θ) / 180. This formula derives from the relationship between an arc and the circumference of a circle.

How do you convert degrees to radians?

To convert degrees to radians, multiply the angle in degrees by π/180. For example, 90 degrees equals π/2 radians (approximately 1.5708). The formula is: radians = degrees × (π/180). Conversely, to convert radians to degrees, multiply by 180/π.

What is the difference between arc length and chord length?

Arc length is the distance along the curved path of the arc, measured along the circumference. Chord length is the straight-line distance between the two endpoints of the arc. Arc length is always greater than or equal to chord length, with equality only when the arc degenerates to a point.

How do you find the radius from arc length and angle?

To find the radius from arc length and angle, use r = s/θ where s is the arc length and θ is the angle in radians. If the angle is in degrees, first convert to radians by multiplying by π/180, then divide the arc length by this value.

What is a sector of a circle?

A sector is a region of a circle bounded by two radii and the arc between them, like a slice of pizza. The central angle determines the size of the sector. A semicircle (180 degrees) and quadrant (90 degrees) are special cases of sectors.

How is arc length used in real life?

Arc length calculations are used in engineering for curved roads and railways, architecture for domes and arches, manufacturing for circular parts and gears, navigation for calculating distances on Earth's surface, and physics for analyzing rotational motion and pendulums.

Related Tools

You might also find these calculators useful:

Additional Resources

Reference this content, page, or tool as:

"Arc Length Calculator" at https://MiniWebtool.com/arc-length-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 15, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Angle Converter
Area of A Sector Calculator
Cosine Calculator
Curl CalculatorNew
Curvature Calculator
Derivative Calculator
Divergence CalculatorNew
Double Integral Calculator
Integral Calculator
Line Integral CalculatorNew

Geometry Calculators:

Top & Updated:

Random PickerRandom Name PickerBatting Average CalculatorLine CounterRelative Standard Deviation CalculatorFPS ConverterSort NumbersERA CalculatorMAC Address GeneratorRemove SpacesInstagram User ID LookupWord to Phone Number ConverterFacebook User ID LookupMAC Address LookupSum CalculatorFeet and Inches to Cm ConverterOPS CalculatorRandom Quote GeneratorSHA256 Hash GeneratorRandom Truth or Dare GeneratorPercent Off CalculatorBitwise CalculatorSquare Root (√) CalculatorVertical Jump CalculatorRoman Numerals ConverterDoubling Time CalculatorSalary Conversion CalculatorLog Base 10 CalculatorSlope and Grade CalculatorNumber of Digits CalculatorMP3 LooperAudio SplitterOn Base Percentage CalculatorSlugging Percentage CalculatorPhone Number ExtractorSaturn Return CalculatorCaffeine Overdose CalculatorMerge VideosSun, Moon & Rising Sign Calculator 🌞🌙✨One Rep Max (1RM) CalculatorClothing Size ConverterRandom IMEI GeneratorCompare Two StringsNumber to Word ConverterAI Text HumanizerFirst n Digits of PiDecimal to BCD ConverterCompound Growth CalculatorCm to Feet and Inches ConverterJob FinderBinary to Gray Code ConverterRandom Poker Hand GeneratorRandom Birthday GeneratorWHIP CalculatorOctal CalculatorRandom Fake Address GeneratorRandom Superpower GeneratorGrade CalculatorAdd Prefix and Suffix to TextRandom Activity GeneratorImage ResizerBCD to Decimal ConverterFile Size ConverterRandom Movie PickerWAR CalculatorYouTube Channel StatisticsVideo to Image ExtractorTime Duration CalculatorPercent Growth Rate CalculatorText FormatterRemove AccentRandom Writing Prompt GeneratorOutlier CalculatorQuotient and Remainder CalculatorStair CalculatorInvisible Text GeneratorLove Compatibility CalculatorCM to Inches ConverterDay of Year CalendarAdd Text to ImageRandom Object GeneratorArc Length CalculatorList of Prime NumbersGray Code to Binary ConverterAI Punctuation AdderRandom Chess Opening GeneratorRemove Leading Trailing SpacesConnect the Dots GeneratorLeap Years ListVideo CropperRandom Loadout GeneratorRandom Number PickerImage SplitterModulo CalculatorImage CompressorList RandomizerNumber ExtractorSHA512 Hash GeneratorTaco Bar CalculatorEmail ExtractorURL ExtractorAI ParaphraserDay of the Year Calculator - What Day of the Year Is It Today?Video CompressorBinary to BCD ConverterIP Address to Hex ConverterSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterLottery Number GeneratorBCD to Hex ConverterMedian CalculatorStandard Error CalculatorBreak Line by CharactersAverage CalculatorPVIFA CalculatorReverse VideoHypotenuse CalculatorRemove Audio from VideoActual Cash Value CalculatorScientific Notation to Decimal ConverterAngel Number CalculatorLog Base 2 CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence Expander📅 Date CalculatorLbs to Kg ConverterHex to Decimal ConverterRandom Group GeneratorConvolution CalculatorMAC Address AnalyzerRandom String GeneratorAmortization CalculatorMarkup CalculatorPVIF CalculatorName Number CalculatorDecimal to Hex ConverterInstagram Font GeneratorSocial Media Image Size GuideTikTok Money CalculatorTwitter/X Character CounterTwitter/X Timestamp ConverterYouTube Watch Time CalculatorTwitch Earnings CalculatorYouTube Shorts Monetization CalculatorFacebook Ad Cost CalculatorSocial Media ROI CalculatorSocial Media Post Time OptimizerSocial Media Username CheckerCTR CalculatorROAS CalculatorInfluencer ROI CalculatorForce CalculatorAcceleration CalculatorVelocity CalculatorMomentum CalculatorProjectile Motion CalculatorKinetic Energy CalculatorPotential Energy CalculatorWork and Power CalculatorDensity CalculatorPressure CalculatorIdeal Gas Law CalculatorFree Fall CalculatorTorque CalculatorHorsepower CalculatorDilution CalculatorChemical Equation BalancerStoichiometry CalculatorPercent Yield CalculatorEmpirical Formula CalculatorBoiling Point CalculatorTitration CalculatorMole/Gram/Particle ConverterIrregular Polygon Area CalculatorFrustum CalculatorTorus Calculator3D Distance CalculatorGreat Circle Distance CalculatorCircumscribed Circle (Circumcircle) CalculatorInscribed Circle (Incircle) CalculatorAngle Bisector CalculatorTangent Line to Circle CalculatorHeron's Formula CalculatorCoordinate Geometry Distance CalculatorVolume of Revolution CalculatorSurface of Revolution CalculatorParametric Curve GrapherRiemann Sum CalculatorTrapezoidal Rule CalculatorSimpson's Rule CalculatorImproper Integral CalculatorL'Hôpital's Rule CalculatorMaclaurin Series CalculatorPower Series CalculatorSeries Convergence Test CalculatorInfinite Series Sum CalculatorAverage Rate of Change CalculatorInstantaneous Rate of Change CalculatorRelated Rates SolverOptimization Calculator (Calculus)Gradient Calculator (Multivariable)Divergence CalculatorCurl CalculatorLine Integral CalculatorSurface Integral CalculatorJacobian Matrix CalculatorNewton's Method CalculatorRREF Calculator (Row Echelon Form)Matrix Inverse CalculatorMatrix Multiplication CalculatorDot Product CalculatorCross Product CalculatorVector Magnitude CalculatorUnit Vector CalculatorAngle Between Vectors CalculatorNull Space CalculatorColumn Space CalculatorCramer's Rule CalculatorMatrix Diagonalization CalculatorQR Decomposition CalculatorCholesky Decomposition CalculatorMatrix Power CalculatorCharacteristic Polynomial CalculatorBayes' Theorem CalculatorF-Test / F-Distribution CalculatorHypergeometric Distribution CalculatorNegative Binomial Distribution CalculatorGeometric Distribution CalculatorExponential Distribution CalculatorWeibull Distribution CalculatorBeta Distribution CalculatorSpearman Rank Correlation CalculatorFisher's Exact Test CalculatorContingency Table CalculatorOdds Ratio CalculatorRelative Risk CalculatorEffect Size CalculatorPermutations with Repetition CalculatorModular Exponentiation CalculatorPrimitive Root CalculatorPerfect Number CheckerAmicable Number CheckerTwin Prime FinderMersenne Prime CheckerGoldbach Conjecture VerifierMöbius Function CalculatorEgyptian Fraction CalculatorFibonacci Number CheckerDigital Root CalculatorPartition Function CalculatorBoolean Algebra SimplifierKarnaugh Map (K-Map) SolverLogic Gate SimulatorGraph Coloring CalculatorTopological Sort CalculatorAdjacency Matrix CalculatorRecurrence Relation SolverInclusion-Exclusion CalculatorLinear Programming SolverTraveling Salesman Solver (TSP)Hamiltonian Path CheckerPlanar Graph CheckerNetwork Flow Calculator (Max Flow)Stable Marriage Problem SolverFirst-Order ODE SolverSecond-Order ODE SolverDirection Field / Slope Field PlotterEuler's Method CalculatorBernoulli ODE SolverSystem of ODEs SolverGroup Theory Order CalculatorRing and Field CalculatorJordan Normal Form CalculatorMatrix Exponential CalculatorTensor Product CalculatorFast Fourier Transform (FFT) CalculatorZ-Transform CalculatorNumerical Integration CalculatorTOML to JSON ConverterJSON to CSV ConverterXML to JSON ConverterSQL to MongoDB Query ConverterCSS Flexbox PlaygroundCSS Grid GeneratorJWT GeneratorBcrypt Hash Generator / CheckerColor Code Converter (All Formats)Git Command Generator.env File GeneratorLorem Picsum / Placeholder Image GeneratorText to Binary/Hex/ASCII ConverterSyllable CounterSentence CounterParagraph CounterSpeaking Time CalculatorReading Time CalculatorWhitespace VisualizerStrikethrough Text GeneratorTorque Converter (Nm, ft-lb, kgf-cm)Data Transfer Rate ConverterFuel Efficiency ConverterAstronomical Unit ConverterRing Size ConverterPaper Size ReferenceGas Mileage CalculatorEV Range CalculatorEV Charging Time Calculator0–60 / Quarter Mile CalculatorCar Lease CalculatorVehicle Towing Capacity CalculatorExposure Triangle CalculatorCrop Factor CalculatorMegapixel to Print Size CalculatorPhoto File Size EstimatorMusic BPM TapperMusic Key TransposerVideo Bitrate CalculatorSeed Germination Rate CalculatorFertilizer Calculator (NPK)Raised Bed Soil CalculatorFrost Date CalculatorLawn Fertilizer CalculatorCompost Calculator (C:N Ratio)Solar Panel CalculatorSolar ROI CalculatorHome Energy Audit CalculatorAppliance Energy Cost CalculatorWater Usage CalculatorElectricity Generation Cost CalculatorHeat Loss CalculatorFlight Distance CalculatorTravel Budget CalculatorJet Lag CalculatorPacking List GeneratorTip Splitter (Advanced)Lease vs Buy CalculatorHourly Rate Calculator (Freelancer)Invoice Late Fee CalculatorESPP CalculatorStock Split CalculatorOptions Probability CalculatorDollar to Gold ConverterBeam Load CalculatorPipe Flow CalculatorBolt Torque CalculatorSteel Weight CalculatorGravel, Sand & Topsoil CalculatorRandom Sentence GeneratorRandom Paragraph GeneratorRandom Math Problem GeneratorRandom Bible Verse GeneratorRandom Cat/Dog Name GeneratorRandom Debate Topic Generator