Arc Length Calculator

Calculate the length of a curve using the arc length formula.

Input Type

Function y = f(x)

From x =

To x =

Use this free arc length calculator to find the distance along any curve. Enter your function in Cartesian or parametric form, set the bounds, and get instant results with step-by-step solutions. Perfect for calculus students and engineers working with curved paths.

What is Arc Length?

Curved architectural structure in grayscale.

Arc length is the distance along a curved line between two points. Unlike straight-line distance, arc length accounts for every twist and turn of the curve. Calculating arc length requires calculus because you’re essentially summing infinitely many tiny straight segments.

The Arc Length Formula

For Cartesian Curves $ y = f(x) $

$$ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$

This formula comes from the Pythagorean theorem applied to infinitesimal segments of the curve. Each tiny piece has horizontal length $ dx $ and vertical length $ dy $, so its length is $ \sqrt{dx^2 + dy^2} $.

For Parametric Curves $ x(t), y(t) $

$$ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt $$

When both $ x $ and $ y $ depend on a parameter $ t $, you compute the derivatives with respect to $ t $ and integrate over the parameter range.

How to Use This Calculator

Choose Input Type – Select Cartesian for functions like $ y = x^2 $ or Parametric for curves defined by $ x(t) $ and $ y(t) $.
Enter Your Functions – Type the function(s) using standard math notation.
Set Bounds – Specify the interval over which to calculate the arc length.

Common Examples

Circle (Parametric)

$ x(t) = r\cos(t) $, $ y(t) = r\sin(t) $ from $ t = 0 $ to $ 2\pi $ gives circumference $ 2\pi r $

Parabola $ y = x^2 $

From $ x = 0 $ to $ x = 1 $, the arc length is approximately 1.4789

Semicircle $ y = \sqrt{1-x^2} $

From $ x = -1 $ to $ x = 1 $, the arc length is $ \pi $ (half the circumference)

Why Arc Length Matters

Engineering Applications

Cable and wire length calculations
Road and railway design
Pipeline routing

Physics Applications

Distance traveled along a curved path
Work done by forces along curves
Center of mass calculations

Computer Graphics

Rendering smooth curves
Animation path length
Font design

Numerical Integration

Most arc length integrals cannot be solved analytically. This calculator uses numerical methods (Simpson’s rule) to approximate the integral. The visualization shows the actual curve being measured.