Arc Length of Parametric Curve Calculator

This calculator finds the exact arc length of a curve defined by parametric equations x(t) and y(t) over a given interval. You enter the two parametric functions, the lower and upper bounds of t, and the calculator returns the total length of the curve in the same units as x and y. The result is computed using the standard integral formula, evaluated numerically with high precision.

Formula

The arc length L of a parametric curve given by x(t) and y(t) from t = a to t = b is:

L = ∫_a^b √( (dx/dt)² + (dy/dt)² ) dt

Where:

• x(t) – the x-coordinate as a function of parameter t (unit: length, e.g., meters or feet)
• y(t) – the y-coordinate as a function of t (unit: length)
• t – the parameter (unitless, often time or angle)
• a, b – lower and upper bounds of t (unitless)
• L – arc length (unit: same as x and y)

The expression under the square root is the magnitude of the velocity vector (dx/dt, dy/dt). Intuitively, you are summing tiny straight-line segments along the curve; as the segments shrink to zero, the sum becomes the integral.

Worked Example

Using the default values from the calculator: x(t) = cos(t), y(t) = sin(t), from t = 0 to t = 2Ï€ (approximately 6.283185).

Step 1: Compute derivatives.
dx/dt = -sin(t), dy/dt = cos(t)

Step 2: Square and add.
(-sin(t))² + (cos(t))² = sin²(t) + cos²(t) = 1

Step 3: Take the square root.
√1 = 1

Step 4: Integrate from 0 to 2Ï€.
∫₀^{2π} 1 dt = [t]₀^{2π} = 2π ≈ 6.2832

The arc length is 6.2832 units, which is the circumference of a unit circle. The calculator returns 6.2832, matching this result.

How the Calculator Works

Because most parametric curves do not have a closed-form antiderivative, the calculator uses numerical integration (adaptive Simpson or Gauss-Kronrod) to approximate the integral to high accuracy. You provide the two functions using standard syntax: use * for multiplication, ^ for exponentiation, and functions like sin, cos, sqrt, exp, ln, etc. For example, to enter x(t) = t², type t^2. The calculator symbolically differentiates your expressions with respect to t, then evaluates the integrand at many points and sums them.

The result is given as a single number with four decimal places. The unit of length is whatever unit you used for x and y; if x and y are in meters, the arc length is in meters.

When to Use This Calculator

Use this calculator whenever a curve is described by parametric equations rather than y = f(x). Common applications include:

• Physics: path length of a projectile or a particle moving along a trajectory
• Engineering: length of a cam profile or a gear tooth
• Computer graphics: length of a Bézier curve or spline
• Mathematics: verifying textbook problems or exploring curves like cycloids, ellipses, or spirals

For a curve given explicitly as y = f(x), you would instead use the formula L = ∫√(1 + (dy/dx)²) dx. The parametric version is more general because it handles curves that loop or go backward in x.

Common Mistakes and Misconceptions

A frequent error is forgetting to square the derivatives. The formula uses the sum of squares, not the sum of the derivatives. Another mistake is using degrees for trigonometric functions; the calculator assumes radians unless you use sind/cosd. Also, ensure the parameter interval [a, b] actually covers the portion of the curve you want; if you enter a larger interval, you get the length of the whole traced path, which may include retracing.

One misconception is that the arc length equals the distance between endpoints. That is only true for a straight line. For any curved path, the arc length is greater than or equal to the straight-line distance. Also, the parameter t does not have to be time; it can be any variable, but the result is independent of how you parameterize the curve (as long as you use the same interval).

How to Choose the Parameter Interval

The interval [a, b] should be the range of t that traces the curve exactly once, unless you intentionally want multiple loops. For a full circle with x = r cos(t), y = r sin(t), use 0 to 2Ï€. For an ellipse x = a cos(t), y = b sin(t), the full perimeter is also 0 to 2Ï€. For a cycloid, one arch is from t = 0 to t = 2Ï€. If you are unsure, plot the curve or think about the period of the trigonometric functions.

Limitations and Accuracy

The calculator gives a numerical approximation, not a symbolic exact answer. For most practical purposes, four decimal places is sufficient. However, if the curve has sharp corners or cusps (where the derivative is undefined), the numerical method may struggle and you might need to split the interval at those points. Also, if the curve is extremely long or the interval is huge, the computation may take longer but still works. The result is always in the same unit as x and y, so if you mix units (e.g., x in meters and y in feet), the result will be meaningless. Always use consistent units.

Frequently Asked Questions

Can I use this for a curve in 3D?
No, this calculator handles only 2D parametric curves. For 3D, you would need an additional z(t) term and the formula L = ∫√(x’² + y’² + z’²) dt.

What if my curve is given in polar coordinates?
You can convert polar to parametric: x = r(θ) cos(θ), y = r(θ) sin(θ), then use this calculator with t = θ. Alternatively, use a dedicated polar arc length calculator.

Why is the result different from the straight-line distance?
Because the arc length follows the actual curve, not a straight line. For a curved path, the arc length is always larger than the Euclidean distance between the endpoints.

Can I enter functions like ln(t) or e^t?
Yes, use ln(t) for natural log and exp(t) or e^t for exponential. The calculator supports standard mathematical functions.

Does the calculator handle piecewise curves?
No, you must split the curve into pieces and sum the lengths manually. For each piece, run the calculator with the appropriate interval and add the results.

What does the result mean if the curve is a closed loop?
If the interval covers the full loop, the result is the perimeter of the loop. For example, a circle of radius r gives 2Ï€r.

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in x | x(t) expression | - | cos(t) | 0
in y | y(t) expression | - | sin(t) | 0
in a | Lower bound t | - | 0 | 0
in b | Upper bound t | - | 6.283185307179586 | 0
out length | Arc length | units | integral(sqrt((diff(x,t))^2 + (diff(y,t))^2), t, a, b) | 4