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:

3D Distance CalculatorNew
Angle Bisector CalculatorNew
Angle Converter
Arccos (Inverse Cosine) Calculator
Arcsin Calculator
Area of A Sector Calculator
Cosine Calculator
Curvature Calculator
Derivative Calculator
Double Integral Calculator

Geometry Calculators:

Top & Updated:

Random PickerRandom Name PickerInstagram User ID LookupFPS ConverterLine CounterRelative Standard Deviation CalculatorSort NumbersImage ResizerMAC Address GeneratorBatting Average CalculatorRemove SpacesFacebook User ID LookupERA CalculatorRandom Truth or Dare GeneratorFeet and Inches to Cm ConverterWord to Phone Number ConverterMAC Address LookupSun, Moon & Rising Sign Calculator 🌞🌙✨Slope and Grade CalculatorPercent Off CalculatorSum CalculatorBitwise CalculatorMP3 LooperAudio SplitterInvisible Text GeneratorRandom IMEI GeneratorRoman Numerals ConverterAI Text HumanizerWord Ladder GeneratorRandom Credit Card GeneratorNumber of Digits CalculatorSHA256 Hash GeneratorHalfway Date Calculator📷 OCR / Image to TextLog Base 10 CalculatorRandom Superpower GeneratorVertical Jump CalculatorRandom Quote GeneratorWAR CalculatorMerge VideosRandom Birthday GeneratorCm to Feet and Inches ConverterPhone Number ExtractorMaster Number CalculatorSalary Conversion CalculatorRandom Fake Address Generator⬛ Aspect Ratio CalculatorOn Base Percentage CalculatorRandom Activity GeneratorSaturn Return CalculatorRandom Poker Hand GeneratorSquare Root (√) CalculatorOPS CalculatorFile Size ConverterSlugging Percentage CalculatorRandom Meal GeneratorRandom Writing Prompt GeneratorLeap Years ListText FormatterCompound Growth CalculatorCaffeine Overdose CalculatorBinary to Gray Code ConverterNumber to Word ConverterIP Subnet CalculatorOctal CalculatorCompare Two StringsRandom Movie PickerLove Compatibility CalculatorBattery Life CalculatorRandom Loadout GeneratorDecimal to BCD ConverterVideo to Image ExtractorBCD to Decimal Converter📅 Date CalculatorMercury Retrograde CalendarStair CalculatorCM to Inches Converter🖱️ Click CounterYouTube Channel StatisticsConnect the Dots GeneratorQuotient and Remainder CalculatorFirst n Digits of PiPER CalculatorAdd Text to ImageName Number CalculatorSHA512 Hash GeneratorPercent Growth Rate Calculator🎰 Gacha Pity CalculatorBingo Card GeneratorAdd Prefix and Suffix to TextImage CompressorRemove AccentOutlier CalculatorWeight Loss CalculatorImage SplitterCoin FlipperVideo CompressorRandom Object GeneratorGray Code to Binary ConverterArc Length CalculatorWhat is my Lucky Number?Day of the Year Calculator - What Day of the Year Is It Today?What is my Zodiac Sign?Lottery Number GeneratorList of Prime NumbersAstrological Element Balance CalculatorSmall Text Generator ⁽ᶜᵒᵖʸ ⁿ ᵖᵃˢᵗᵉ⁾Flip VideoDiff CheckerTime Duration CalculatorURL ExtractorAngel Number CalculatorIP Address to Hex ConverterAcreage CalculatorBcrypt Hash Generator / CheckerRandom Line PickerBreak Line by CharactersRemove Leading Trailing SpacesMultiple Fraction CalculatorNumber ExtractorRandom Time GeneratorHeight Percentile CalculatorAI Language DetectorBinary to BCD ConverterProportion CalculatorMartingale Strategy CalculatorModulo CalculatorWHIP CalculatorDMS to Decimal Degrees ConverterAI Punctuation AdderRandom Chess Opening GeneratorWord Scramble GeneratorPercentile CalculatorCone Flat Pattern (Template) GeneratorDay of Year CalendarRandom Number PickerVideo SplitterLife Path Number CalculatorLong Division CalculatorMandelbrot Set ExplorerRandom Emoji GeneratorPVIFA CalculatorLongest Day of the YearAdjust Video SpeedPVIF CalculatorRandom Chord GeneratorAntilog CalculatorMorse Code GeneratorYouTube Tag ExtractorCollage MakerRatio to Percentage CalculatorBoxing Punch Power CalculatorHypotenuse CalculatorEmail ExtractorSteel Weight CalculatorCrossword Puzzle MakerSum of Positive Integers CalculatorBonus CalculatorRandom User-Agent GeneratorDestiny Number CalculatorFirst n Digits of eAI ParaphraserLunar Calendar ConverterColor InverterPercent to PPM ConverterRandomize NumbersShort Selling Profit CalculatorGrade CalculatorMD5 Hash GeneratorMiter Angle CalculatorSquare Numbers ListFraction CalculatorList RandomizerHTML Compressor🔍 Plagiarism CheckerRounding CalculatorArctan2 CalculatorIs it a Prime Number?Multiplication CalculatorName RandomizerBirth Day of the Week CalculatorMolarity CalculatorTrigonometric Equation SolverDue Date CalculatorExponential Decay CalculatorRandom Group GeneratorYouTube Earnings EstimatorHebrew Calendar ConverterRandom Name GeneratorMAC Address AnalyzerPizza Value CalculatorReverse TextBoiling Point CalculatorNumber RandomizerRemove Line BreaksRock Paper Scissors GeneratorLog (Logarithm) Calculator🎲 Loot Drop Probability CalculatorSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterBCD to Hex ConverterMedian CalculatorStandard Error CalculatorAverage CalculatorReverse VideoRemove Audio from VideoActual Cash Value CalculatorScientific Notation to Decimal ConverterLog Base 2 CalculatorRoot Mean Square CalculatorSHA3-256 Hash GeneratorAI Sentence ExpanderLbs to Kg ConverterHex to Decimal ConverterConvolution CalculatorRandom String GeneratorAmortization CalculatorMarkup 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 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 CalculatorROT13 Encoder/DecoderAtbash Cipher ToolVigenère Cipher ToolPronunciation IPA ConverterHemingway-Style Readability EditorSentence Length Variance AnalyzerWord Frequency AnalyzerBusiness Days CalculatorAdd Business Days to DateDate Pattern GeneratorHow Long Until CalculatorHow Long Ago CalculatorBirthday Across Cultures CalculatorHijri Calendar ConverterInsulin Sensitivity Factor CalculatorCarb-to-Insulin Ratio CalculatorLean Body Mass to Strength CalculatorOne-Mile Walk Test (Rockport) CalculatorCooper 12-Minute Run CalculatorFFMI CalculatorAPGAR Score CalculatorGlasgow Coma Scale CalculatorWells Score Calculator (DVT/PE)Tennis Score TrackerSoccer xG (Expected Goals) CalculatorCricket Run Rate CalculatorRugby Points CalculatorRace Time PredictorSwimming SWOLF CalculatorYoga Pose Hold TimerFishing Knot Strength CalculatorBike Gear Ratio CalculatorClimbing Grade ConverterWine Pairing SuggesterStandard Drink CalculatorCaffeine Half-Life TrackerSpice Substitution FinderDietary Restriction Recipe FilterMarinade Time CalculatorFermentation Time CalculatorSmoking Wood Pairing GuideFreelance Project Pricing CalculatorSaaS Pricing CalculatorSubscription Cost TrackerSide Hustle ROI CalculatorRemote Work Savings CalculatorCoffee Habit Cost CalculatorGym vs Home Workout Cost CalculatorLunch Cost CalculatorWealth Growth Visualizer1031 Exchange CalculatorRental Yield CalculatorCash-on-Cash Return CalculatorBRRRR Method CalculatorSection 8 Rent CalculatorRoommate Rent SplitterAirbnb Pricing OptimizerStatute of Limitations CalculatorSentence Reduction CalculatorSales Tax Nexus CheckerPatent Filing Fee CalculatorTrademark Class FinderWill Asset Distribution CalculatorWorkers' Compensation CalculatorStopping Distance CalculatorTrip Cost SplitterVehicle Weight Distribution CalculatorTrailer Tongue Weight CalculatorTire Tread Wear CalculatorEngine Compression Ratio CalculatorHeadlight Beam Distance CalculatorCat Litter Box CalculatorAquarium Heater Wattage CalculatorBird Cage Size CalculatorReptile Habitat UVB CalculatorPet Travel Crate Size FinderHorse Hay CalculatorCrochet Hook Size ConverterKnitting Needle Size ConverterKnitting Pattern CalculatorCross-Stitch Floss CalculatorQuilt Binding CalculatorOrigami Paper Size CalculatorPottery Clay Shrinkage CalculatorBeading Pattern CalculatorResin Casting Volume CalculatorEmbroidery Thread Length CalculatorHiking Pace Calculator (Naismith's Rule)Backpacking Food Weight CalculatorTent Footprint Size CalculatorSleeping Bag Temperature Rating GuideKnot Tying Reference ToolStar Visibility CalculatorTide Time CalculatorSun Position CalculatorReynolds Number CalculatorBernoulli Equation CalculatorHeat Transfer CalculatorThermal Expansion CalculatorSpecific Heat Capacity CalculatorGear Ratio Calculator (Mechanical)Pulley System CalculatorHydraulic Cylinder Force CalculatorBelt Length CalculatorCloset Capsule CalculatorStorage Unit Size CalculatorMoving Box Quantity CalculatorGift Card Tip CalculatorGas vs Electric Cost ComparisonPrint Cost CalculatorHair Dye Mixing CalculatorLaundry Detergent Dosage Calculator