Calculate and visualize arc length of curves with the integral formula. Supports Cartesian, parametric, and polar curves.
Arc length is the distance you would actually travel along a curve, not the straight-line distance between its endpoints. This tool evaluates the arc-length integral - sqrt(1 + f'(x)^2) for a Cartesian graph y = f(x), with matching forms for parametric and polar curves - and draws the curve with the measured arc highlighted. You supply the equation and the two bounds; the page returns the length as a number. Most users' first surprise is that the result always exceeds the endpoint-to-endpoint distance, since every bend in the curve adds length.
A concrete input and expected output from the current implementation.
Input
y = x^2, from x = 0 to x = 1
Expected output
Arc length ~ 1.4789 (closed form: (2*sqrt(5) + ln(2 + sqrt(5))) / 4)
The integrand is sqrt(1 + (2x)^2) = sqrt(1 + 4x^2); integrating it from 0 to 1 gives (2*sqrt(5) + ln(2 + sqrt(5)))/4 = 1.47894... Since the curve climbs from (0,0) to (1,1), the arc is longer than the sqrt(2) ~ 1.4142 chord between those points.
The tool builds the integrand from what you type: for y = f(x) it is sqrt(1 + (dy/dx)^2); for a parametric curve (x(t), y(t)) it is sqrt((dx/dt)^2 + (dy/dt)^2); for polar r(theta) it is sqrt(r^2 + (dr/dtheta)^2). Each mode expects different inputs - an x-interval, a parameter range, or an angle range - and the integrand always involves the derivative of what you entered.
The two numbers you enter are integration limits, and only the portion of the curve between them is measured and highlighted on the plot. Edit a bound and the highlight and the reported number change together. If the arc you intend is not the arc that is highlighted, you are measuring a different segment; check the drawing first when a result looks wrong.
The straight line is the shortest path between two points, and your curve is not straight. The integral adds up infinitesimal steps taken along the curve itself, so every bend and bulge contributes. The two agree only when the segment is perfectly straight. For example, y = x^2 from 0 to 1 measures about 1.479, versus sqrt(2) ~ 1.4142 for the chord.
No, not directly. The arc-length formula needs a derivative, and a point cloud or image has no derivative until you fit a smooth curve to it. Fit a function to your data first and enter that equation; the length you get is then only as accurate as the fit, so smooth, closely spaced data gives the most trustworthy result.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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