All calculatorsArc Length Calculator
Calculate the length of a curve using Circle arc (fast), Cartesian (y = f(x)), Parametric (x(t), y(t)), or Polar (r(θ)). Results are unit-aware and include optional step-by-step.
Background
Arc length measures “how long the curve is” along its path. For many functions, the arc length integral doesn’t simplify nicely, so we approximate it with high-accuracy numerical integration.
How to use this calculator
- Start with Circle arc if you have a radius and central angle.
- For equations, choose Cartesian / Parametric / Polar and enter the interval.
- Pick units, then click Calculate.
How this calculator works
- Circle arc: converts θ to radians and computes L = r·θ.
- Cartesian: numerically evaluates ∫ √(1+(dy/dx)²) dx.
- Parametric: numerically evaluates ∫ √((dx/dt)²+(dy/dt)²) dt.
- Polar: numerically evaluates ∫ √(r²+(dr/dθ)²) dθ.
Formula & Equation Used
Circle arc: L = r·θ (θ in radians)
Cartesian: L = ∫ab √(1+(dy/dx)²) dx
Parametric: L = ∫ √((dx/dt)²+(dy/dt)²) dt
Polar: L = ∫ √(r²+(dr/dθ)²) dθ
Example Problems & Step-by-Step Solutions
Example 1 — Circle arc (degrees)
Radius r = 3 m, angle θ = 60°.
- Convert θ to radians: 60° = π/3 ≈ 1.0472 rad.
- Compute L = r·θ = 3 × 1.0472 ≈ 3.1416 m.
Example 2 — Circle arc (radians)
Radius r = 10 cm, angle θ = 1.2 rad.
- No conversion needed (already radians).
- Compute L = r·θ = 10 × 1.2 = 12 cm.
Example 3 — Cartesian curve
y = sin(x) on [0, π].
- Compute derivative: dy/dx = cos(x).
- Set up arc length: L = ∫₀^π √(1 + cos²(x)) dx.
- This integral is evaluated numerically by the calculator.
Example 4 — Parametric circle
x(t)=cos(t), y(t)=sin(t) on [0, 2π].
- Compute derivatives: dx/dt = −sin(t), dy/dt = cos(t).
- Speed: √((dx/dt)²+(dy/dt)²) = √(sin²(t)+cos²(t)) = 1.
- Arc length: L = ∫₀^{2π} 1 dt = 2π.
Frequently Asked Questions
Q: Do I need degrees or radians?
For Circle arc and Polar, you can choose. Internally, the calculator converts to radians.
Q: Why do some arc lengths use numerical integration?
Many curves don’t have a simple closed-form arc length. Numerical integration gives an accurate approximation.
Q: What if my function is undefined on part of the interval?
The calculator may error out. Try restricting the interval to where the curve is defined.
Q: Can arc length be smaller than the straight-line distance?
No — the straight-line distance (chord) is always the shortest path between two points.
Q: What does “High accuracy” change?
It tightens the integration tolerance and recursion depth for more precision (slightly slower).
