Arc Length Calculator Parametric

This calculator finds the length of a curve defined by parametric equations x(t) and y(t) over a given interval. You enter the two parametric functions, the start and end values of the parameter t, and the tool computes the arc length using the standard integral formula. The result is given in the same units as the coordinate axes (e.g., meters, feet, or any consistent unit).

Formula

For a parametric curve given by x = f(t) and y = g(t), the arc length from t = a to t = b is:

L = ∫_a^b √[(dx/dt)² + (dy/dt)²] dt

Where:

• L = arc length (same unit as x and y)
• t = parameter (often time, angle, or a dimensionless variable)
• dx/dt = derivative of x with respect to t
• dy/dt = derivative of y with respect to t
• a, b = lower and upper bounds of t

This formula is the parametric version of the Pythagorean distance integrated over the curve. It works for any smooth curve (where the derivatives exist and are continuous).

Worked Example

Using the default values: x(t) = t², y(t) = t³/3, from t = 0 to t = 2.

1. Compute derivatives: dx/dt = 2t, dy/dt = t².
2. Square them: (2t)² = 4t², (t²)² = t�.
3. Sum: 4t² + t� = t²(4 + t²).
4. Integrate: L = ∫₀² √[t²(4 + t²)] dt = ∫₀² t√(4 + t²) dt.
5. Substitute u = 4 + t², du = 2t dt → (1/2)∫₄� √u du = (1/2) * (2/3)u^(3/2) from 4 to 8.
6. Evaluate: (1/3)(8^(3/2) – 4^(3/2)) = (1/3)(22.627 – 8) = 4.876.

The calculator gives approximately 4.876 units, matching the manual calculation.

When to Use Parametric Arc Length

Use this calculator when your curve is expressed as x(t) and y(t) rather than y = f(x). Common situations include:

• Physics: trajectory of a projectile (x and y as functions of time)
• Engineering: path of a robot arm or cam profile
• Computer graphics: splines and Bezier curves
• Astronomy: orbits described parametrically

If your curve is given as y = f(x), you can convert it to parametric form by setting x = t and y = f(t), or use a standard Cartesian arc length calculator.

How to Read the Result

The output is the total length of the curve along its path, not the straight-line distance between endpoints. For example, a semicircle of radius 1 from t=0 to t=π (x = cos t, y = sin t) gives arc length π ≈ 3.14159, while the chord length is 2. The arc length is always greater than or equal to the straight-line distance.

Units: if x and y are in meters, the arc length is in meters. If they are in feet, the result is in feet. The parameter t does not affect the unit.

Common Mistakes and Limitations

• Forgetting to square the derivatives – the formula uses the square root of the sum of squares, not the sum of the derivatives.
• Using degrees for trigonometric functions – if your parametric functions involve sin or cos, ensure you are consistent. Most math contexts use radians, but if you use degrees, the derivatives will be off by a factor of Ï€/180.
• Ignoring absolute value – the integrand is always non-negative, so no absolute value is needed, but be careful when substituting variables.
• Curves that cross themselves – the formula gives the length of the path, not the net displacement. For a loop, you get the total length of the loop, not the distance from start to end.
• Non-smooth points – if the curve has a cusp or corner (e.g., x = t², y = t³ at t=0), the derivative may be zero, but the formula still works if you split the integral at the cusp.

Alternatives and Related Calculations

If your curve is in polar coordinates (r = f(θ)), use the polar arc length formula: L = ∫ √[r² + (dr/dθ)²] dθ. For a 3D parametric curve (x(t), y(t), z(t)), extend the formula to L = ∫ √[(dx/dt)² + (dy/dt)² + (dz/dt)²] dt. Many calculators also offer numerical integration for functions that lack elementary antiderivatives.

Frequently Asked Questions

Can I use this for a curve given as y = f(x)?

Yes, set x = t and y = f(t). For example, y = x² becomes x = t, y = t². Then the calculator will compute the same result as the Cartesian formula.

What if my curve is in 3D?

This calculator handles 2D only. For 3D, you need a formula with three components. You could use a 3D parametric arc length calculator or compute the integral manually.

Why is the result different from the distance between endpoints?

Because the arc length follows the curve, not a straight line. For a curved path, the length is always greater than the straight-line distance between the start and end points.

Can I use this for a closed curve (like a circle)?

Yes. For a circle of radius r, use x = r cos t, y = r sin t, from t = 0 to 2Ï€. The result will be 2Ï€r, the circumference.

What if the integral has no closed form?

The calculator uses numerical integration (like Simpson’s rule or adaptive quadrature) to approximate the value. You can increase the number of intervals for more precision, but the default is usually accurate to several decimal places.

Do I need to enter derivatives?

No. The calculator automatically differentiates your x(t) and y(t) expressions symbolically (or numerically) and then integrates. You only provide the functions and the t-range.

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in t_start | t start | - | 0 | 0
in t_end | t end | - | 2 | 0
in x_expr | x(t) | - | t^2 | 0
in y_expr | y(t) | - | t^3 / 3 | 0
out arc_len | Arc length | units | integral(sqrt((diff(x_expr,t))^2 + (diff(y_expr,t))^2), t, t_start, t_end) | 4