Arc Length Calculator

Measure the exact length of any curve between two points

The arc length of a curve y = f(x) from x = a to x = b is:

L = \int_a^b \sqrt{1 + [f'(x)]^2}\, dx

For parametric curves (x(t), y(t)):

L = \int_{t_1}^{t_2} \sqrt{[x'(t)]^2 + [y'(t)]^2}\, dt

This arc length calculator with graph highlights the curve segment on an interactive graph — see exactly which part of the curve is being measured. Type any function and a range, and the AI computes the exact arc length using numerical integration. Zoom, pan, and explore the curve visually. Works with polynomials, trigonometric, exponential, and parametric curves.

Related Topics

Integrals (lesson)Area Between Curves CalculatorTangent Line Calculator

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What is arc length?
Arc length is the distance along a curve between two points — if you "straightened out" the curve, that would be its arc length. It is always greater than or equal to the straight-line distance between the endpoints.
How is arc length calculated?
The arc length formula integrates \sqrt{1 + [f'(x)]^2} from a to b. Intuitively, at each tiny segment dx, the curve rises by f'(x)·dx, so the segment length is \sqrt{dx^2 + dy^2}. Summing these gives the total arc length.
Can I compute arc length of a parametric curve?
Yes. Type a parametric curve like "(cos(t), sin(t))" and a range like "t from 0 to 2pi". The formula uses \sqrt{(dx/dt)^2 + (dy/dt)^2} instead.
What if no bounds are given?
Arc length requires bounds — without them, the length is infinite for any non-constant function. Specify "from a to b" or the calculator uses the current viewport range.
How accurate is the calculation?
The calculator uses Simpson's rule with 500 subintervals, giving accuracy to about 6 decimal places for smooth functions. This is more than sufficient for all practical and educational purposes.
Why use an arc length calculator with a graph?
The graph shows exactly which part of the curve is being measured. You can see the curve, the endpoints, and understand visually why a curve is longer than the straight-line distance between its endpoints. Interactive zoom and pan let you explore the curve's shape in detail.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.