Arc Length Calculator

“Arc length” means two different things depending on the shape, and this calculator handles both.

Circular arc. For a slice of a circle, arc length is simply s = rθ, where r is the radius and θ is the central angle in radians. If your angle is in degrees, convert first (θ in radians = θ° × π / 180) or use the degree form directly: s = (θ / 360) × 2πr. One radian is, by definition, the angle that makes the arc equal to the radius, which is why the radian form has no clutter. A full turn (θ = 2π) gives s = 2πr, the circumference, so the formula stays consistent with what you already know. This is the version you want for road and railway curves, belt and pulley lengths, gear teeth, and clock-hand travel.

Curve given by a function. For a smooth curve y = f(x) between x = a and x = b there is no shortcut, so you integrate: L = ∫ from a to b of √(1 + (f’(x))²) dx. The idea is Pythagoras applied to infinitely thin segments: over a tiny step dx the curve rises by dy, so the true length of that segment is √(dx² + dy²) = √(1 + (dy/dx)²) dx. A flat line has f’(x) = 0, so the integrand is 1 and the arc length equals the plain width b − a, the shortest any curve can be. The more the graph bends, the more arc length exceeds that width. This calculator evaluates the integral with Simpson’s rule over 1000 steps, far more precise than the inputs ever need.

Two related forms show up often enough to keep handy. A curve written parametrically as x(t), y(t) has L = ∫ √((dx/dt)² + (dy/dt)²) dt, and in polar coordinates r(θ) it becomes L = ∫ √(r² + (dr/dθ)²) dθ. The parametric form is the natural one for circles, ellipses, spirals, and any path that loops back on itself.

One thing students mix up: arc length is not the area under the curve. Area uses f(x); arc length uses √(1 + f’(x)²). They look alike and measure entirely different things.