Starlight Tools

Arc Length Calculator — Circle, Function, Parametric & Polar Curves

Choose Circular Arc when you know a circle’s radius or diameter and central angle: s=rθ. For a calculus curve, choose Function y=f(x), Parametric x(t), y(t), or Polar r(θ). Get numbered steps, an exact result for recognized cases, a decimal approximation, and a graph of your selected segment.

Enter a curve

Curve type

Use ^ for powers and radians for trigonometry. Functions include sin, cos, tan, exp, ln, sqrt, and abs.

Expression syntax and keypad

Select an expression field, then insert a symbol. Use 2*x or 2x, x^2, sqrt(x), 1/3, pi/2, or e. Use parentheses around function arguments. Trigonometric expressions use radians.

Accuracy and display

Tolerance controls numerical subdivision, not guaranteed correct digits. Graph density only changes the plot and CSV. Exact results bypass quadrature. Integration status and estimated error appear with the result.

The calculator labels the result; it does not convert coordinates.

Try an example

Press Ctrl/⌘ + Enter to calculate.

Arc length result

Calculated arc length
—
Enter a curve and bounds to calculate its length.
The arc-length integral will appear here.

Step-by-step solution

    Exact or numerical status will appear after calculation.

    Which arc length formula should I use?

    Mode and required inputsFormulaExample
    Circular: radius r (or diameter/2), central angle θs=rθ; θ in radiansr=3, θ=π/3 → s=π
    Function: y=f(x), bounds a,bL=∫ₐᵇ√(1+[f′(x)]²) dxy=x² on [0,1]
    Parametric: x(t), y(t), bounds a,bL=∫ₐᵇ√([x′(t)]²+[y′(t)]²) dtx=cos(t), y=sin(t) on [0,2π]
    Polar: r(θ), radian bounds α,βL=∫ᵅᵝ√(r²+[r′(θ)]²) dθr=1−cos(θ) on [0,2π]

    Different quantities: Arc or curve length follows the curve. A chord is the straight line between endpoints; for a circle its length is 2r|sin(θ/2)|. Circumference is the complete circle’s length, 2πr. Radius of curvature describes local bending; it is not a traveled distance.

    Calculus formulas assume a sufficiently smooth curve. Split piecewise-smooth curves at corners or singularities. Repeated traversals count repeatedly. Graph arrows follow the entered bounds, even when the length integral reorders them.

    References: OpenStax Calculus Volume 2 §2.4: function arc length, §7.2: parametric curves, and §7.4: polar arc length.

    Curve graph

    Calculate a curve to draw the traced segment.

    Speed / integrand plot

    Horizontal axis: curve parameter. Vertical axis: speed. Unlike the curve graph, these axes use independent scales.

    Accessible sample table
    Five points along the entered traversal; the CSV includes all plotted samples.
    PositionParameterxySpeed

    Calculator methodology and limits

    Publisher
    Starlight Robotics. No independently verified individual mathematics reviewer is credited.
    Calculation checks updated
    Method
    The local DerivativeEngine parser builds an expression tree without executing user code, then differentiates symbolically. Circle mode uses s=rθ. Curve modes scan 1,025 points, try recognized exact identities, and otherwise use 32 initial panels with adaptive Simpson subdivision (maximum depth 16 per panel and 150,000 integration evaluations).
    Accuracy
    Automatic tolerance is 1e-9; manual tolerance ranges from 1e-12 to 0.001. Each initial panel gets 1/32 of the tolerance with local relative scaling. Estimated subdivision error is not a guaranteed bound on total error. Highly oscillatory curves or narrow singularities can escape sampling. Split difficult intervals and compare tighter tolerances.
    Input limits
    Real-valued expressions of at most 300 characters and finite bounds between −10¹² and 10¹². The curve must be smooth throughout the interval.
    Reproducible checks
    Load the worked examples: y=x on [0,1] gives √2, the unit circle gives 2π, and one cycloid arch and the cardioid r=1−cos(θ) each give 8. Parabola and spiral results are checked against their asinh antiderivatives; numerical sine and ellipse results are checked independently.
    Exact-answer policy
    Recognized cases include constant speeds, y=x², the unit circle parameterization, one standard cycloid arch, constant polar radii, r=θ, and one full r=1−cos(θ) cardioid. The engine is not a general symbolic integrator. Decimal outputs use floating-point arithmetic; an exact expression does not make its decimal approximation exact.
    Privacy
    Your equations, bounds, graph, copied summary, and downloaded samples remain on your device.

    How to use the arc length calculator

    1. Choose Circular Arc for radius and angle, Function for y=f(x), Parametric for x(t), y(t), or Polar for r(θ).
    2. Enter the known circle measurements or curve expressions. Use ^ for powers; calculus trigonometry uses radians.
    3. For curves, enter finite bounds, including pi, e, fractions or arithmetic. Reversed bounds give the same nonnegative length.
    4. Calculate and inspect the numbered steps, exact or numerical answer, integration status and graph. Copy results or download samples.

    How to interpret the result

    Length is not signed area

    The square-root integrand is a speed and is nonnegative. Traversing a curve backward does not make its geometric length negative.

    Function mode

    For y=f(x), each small horizontal change dx combines with the vertical change f′(x)dx through the Pythagorean theorem.

    Parametric mode

    The vector velocity is (x′(t),y′(t)). Its magnitude is integrated over the parameter interval, so the parameter itself need not measure distance.

    Corners and singularities

    The numerical method expects smooth input. Split a piecewise-smooth curve at corners, cusps, or singular points, calculate each smooth segment, and add the lengths.

    Worked examples

    Line: y=x, 0≤x≤1

    f′=1; speed=√(1+1²)=√2. Integrate constant speed: L=[√2 x]₀¹=√2 ≈ 1.4142135624.

    Parabola: y=x², 0≤x≤1

    f′=2x; speed=√(1+4x²). Use F(x)=x√(1+4x²)/2+asinh(2x)/4. Thus L=F(1)−F(0)=√5/2+asinh(2)/4 ≈ 1.4789428575.

    Sine: y=sin(x), 0≤x≤π

    f′=cos(x); speed=√(1+cos²(x)). This integral has no elementary antiderivative. Adaptive Simpson integration gives L ≈ 3.8201977890.

    Parametric unit circle, 0≤t≤2π

    x=cos(t), y=sin(t); x′=−sin(t), y′=cos(t). Speed=√(sin²(t)+cos²(t))=1. L=[t]₀²π=2π ≈ 6.2831853072.

    Ellipse: x=3cos(t), y=2sin(t), 0≤t≤2π

    x′=−3sin(t), y′=2cos(t); speed=√(9sin²(t)+4cos²(t)). The complete perimeter needs an elliptic integral; adaptive Simpson integration gives L ≈ 15.8654395893.

    Cycloid: x=t−sin(t), y=1−cos(t), 0≤t≤2π

    x′=1−cos(t), y′=sin(t); speed=2sin(t/2) on this interval. L=[−4cos(t/2)]₀²π=8 (decimal 8.0000000000). The endpoint speeds are zero.

    Circular arc: radius 3, angle 60°

    Convert θ=60π/180=π/3 radians. No differentiation is needed: s=rθ=3(π/3)=π ≈ 3.1415926536.

    Polar circle: r=1, 0≤θ≤2π

    r′=0; speed=√(1²+0²)=1. L=[θ]₀²π=2π ≈ 6.2831853072.

    Polar cardioid: r=1−cos(θ), 0≤θ≤2π

    r′=sin(θ). Speed=√((1−cos(θ))²+sin²(θ))=2sin(θ/2). Integrate: L=[−4cos(θ/2)]₀²π=8 (decimal 8.0000000000).

    Archimedean spiral: r=θ, 0≤θ≤2π

    r′=1; speed=√(1+θ²). Use F(θ)=(θ√(1+θ²)+asinh(θ))/2. L=π√(1+4π²)+asinh(2π)/2 ≈ 21.2562941482.

    Arc length calculator FAQs

    How do I find circular arc length from radius and angle?

    Choose Circular Arc, enter radius r (or diameter d, giving r=d/2) and the central angle. Use s=rθ with θ in radians, or s=πrα/180 with α in degrees. You can also solve r=s/θ or θ=s/r when the other two values are known.

    Should my angle be in degrees or radians?

    Circle mode accepts either using its angle-unit selector. Function, parametric and polar trigonometry always uses radians. Convert degrees to radians by multiplying by pi/180.

    What is the difference between arc length and chord length?

    Arc length follows the circle: s=rθ. Chord length crosses directly between its endpoints: c=2r|sin(θ/2)|. Except for zero angle, the curved distance is longer than the chord.

    How do I calculate polar arc length?

    Choose Polar and enter r(theta), with bounds in radians. Differentiate r with respect to θ and integrate √(r²+[r′(θ)]²). For r=1−cos(θ) from 0 to 2pi, the exact length is 8.

    Can I get an exact answer?

    Yes for recognized cases, including a straight line, y=x², a unit circle, one standard cycloid arch and selected polar curves. Exact expressions are shown above a decimal approximation. Other inputs use numerical integration; failure to find a closed form is not a proof that none exists.

    Why do many arc-length integrals need numerical methods?

    Squaring derivatives and taking a square root often produces an integrand with no elementary antiderivative. For example, sine-curve lengths and general ellipse perimeters involve elliptic integrals. Adaptive Simpson integration gives a numerical approximation over finite bounds.

    How do I enter π, fractions or symbolic bounds?

    Type pi or π, pi/2, 2*pi, e, 1/3 or simple arithmetic such as (1+2)/4. Bounds must evaluate to real finite constants. The result displays normalized bounds alongside their decimal values.

    What is the arc length formula for y=f(x)?

    For a continuously differentiable function on [a,b], use L=∫ₐᵇ√(1+[f′(x)]²)dx. The 1 represents the horizontal component of each small displacement.

    How do I calculate the length of a parametric curve?

    Differentiate both coordinates with respect to the same parameter, find the speed √([x′(t)]²+[y′(t)]²), and integrate that speed across the parameter interval.

    Why is the result still positive when I reverse the bounds?

    Arc length is distance, not an oriented integral. Reversing the bounds changes the direction of traversal but not the distance traveled, so this calculator orders the bounds before integrating.

    Does a parametric curve that retraces itself count twice?

    Yes. The speed integral measures distance traveled along the parameterization. If the parameterization traces a segment twice, both traversals contribute to the result.

    Can I use degrees for sine and cosine?

    Calculus sine and cosine expressions require radians. Convert degrees to radians before entering a parameter interval. Circular Arc mode has a separate selector that accepts degrees directly.

    What if the curve has a corner or cusp?

    Split the interval at each nonsmooth point, calculate the smooth pieces separately, and add their nonnegative lengths. The calculator reports derivatives that are undefined or extreme inside a single interval.

    Is the numerical error estimate guaranteed?

    No. It estimates the remaining adaptive Simpson quadrature error based on differences between successive subdivisions. It can be less reliable for badly behaved or insufficiently resolved curves, which is why the calculator also scans for non-real values and extreme derivatives.

    Does my curve data leave the browser?

    No. All parsing, differentiation, integration, graphing, copying, and file preparation happen locally. A permalink contains only the inputs you explicitly choose to put in the URL.

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