Arc length

The formula

The core idea is to chop the curve into pieces so short each one is basically a straight line, add up the little hypotenuses, and take the limit. That limit is an integral.

s = ∫ₐᵇ √( 1 + (dy/dx)² ) dx        (curve given as y = f(x))

Reads as: at every x, the curve rises by dy while it runs by dx, so the local step length is √(dx² + dy²). Factor out dx and integrate.

L = ∫ₐᵇ √( (dx/dt)² + (dy/dt)² ) dt   (parametric, x(t) and y(t))

Reads as: speed is the length of the velocity vector; integrate speed over time and you get distance. This is the form a CNC controller and a robot path planner actually use.

L = ∫ₐᵇ √( (dx/dt)² + (dy/dt)² + (dz/dt)² ) dt   (3D, add the z term)

Reads as: same thing, one more axis. Cable in a tray, a helix, a five-axis toolpath.

s = ∫ √( r² + (dr/dθ)² ) dθ          (polar, r = f(θ))

Reads as: as the angle sweeps, you move r·dθ around and dr outward; combine them.

Two closed forms worth memorizing, because they're the ones that actually have one:

Circle circumference:  C = 2πr = πd
Cycloid arch length:   L = 8a   (a = rolling-circle radius)

Where you meet it

Belt and cable length on the bench. A belt wrapping two pulleys isn't 2 × center distance. It's two straight tangent spans plus two circular arcs, and each arc length is r · (wrap angle in radians). Get the wrap angle wrong and your belt is either too short to install or slaps loose. Same math sizes a cable run that curves through a tray instead of going point to point.
CNC feed distance and cycle time. Feed rate is inches per minute along the path. To estimate cycle time or program a constant surface speed, the controller integrates arc length over the toolpath. A contour milled as hundreds of tiny line segments (G01) has a slightly shorter total length than the smooth arc it approximates — that error shows up as a feed-rate and time discrepancy on a curvy part.
Conductor sag and catenary at the test stand. A cable strung between two supports hangs as a catenary, and how much conductor you need to order is its arc length, always longer than the horizontal span. Under-order and you're splicing.
Review board, tolerance stack on a curved feature. When somebody asks "how long is the actual seal groove around that oval flange," the answer is an ellipse perimeter — and that one does not have a clean formula. Knowing that up front keeps you from chasing an exact number that doesn't exist.

How it works

The whole subject is one move: approximate the curve by a chain of straight chords, sum their lengths, and refine. Formally that's Σ √(Δx² + Δy²) going to an integral. Everything else is bookkeeping about how the curve is described — as y(x), as (x(t), y(t)), or in polar.

The gotcha that bites engineers is that most curves have no elementary closed-form arc length. The integrand carries a square root, and square roots of polynomials rarely integrate to anything you can write with sin, log, and friends. The parabola works out (y = x² from 0 to 1 is 1.4789..., and there's a tidy asinh formula for it). The circle and the cycloid work out. The ellipse does not — its perimeter is an elliptic integral, and that entire branch of math is named after this exact failure. So when you need an oval's perimeter, you either integrate numerically or use Ramanujan's approximation:

P ≈ π(a + b)·( 1 + 3h / (10 + √(4 − 3h)) ),   h = ((a − b)/(a + b))²

For a 2×1 ellipse (a=2, b=1) the true perimeter is 9.68844822; Ramanujan gives 9.68844821 — eight good digits, no integral. That's usually more than enough for ordering material.

The mistake people make: treating a curved length as its straight-line chord, or worse, its bounding box. The straight distance between the ends of a cycloid arch is 2πa ≈ 6.28a; the actual arc is 8a. That's a 27% under-count. Belt length, cable pull, weld bead on a curved seam, toolpath time — approximate a curve as a straight line and you are always short, never long, because the straight line is the shortest path between two points by definition.

Second gotcha: units and radians. The r·θ arc-length shortcut only works with θ in radians. Plug in degrees and you're off by 180/π ≈ 57×. It's the single most common arc-length error on a shop floor.

A sanity check you can always run: the arc length between two points is never less than the straight-line distance between them — that lower bound holds for any curve, period. For an upper bound, be careful: only if the curve runs monotonically in both x and y — no switchbacks, no doubling back — is its length at most the width plus the height of its bounding box. A curve that turns around (a full circle is 2πr ≈ 6.28r against extents summing to 4r) blows past that. So split the curve at its turning points, apply the width-plus-height bound to each monotone piece, and sum. If your number falls outside that window, something's wrong.

History

For two thousand years, "rectification" — straightening a curve out into an equal-length line segment — was considered maybe impossible for most curves. Aristotle's followers doubted a curve and a straight line could even have the same length. The circle was the exception everyone knew, and that only in terms of π, which nobody could pin down exactly.

The dam broke in the 1650s. The logarithmic spiral fell first — usually credited to Evangelista Torricelli around 1645, though some sources give John Wallis in the 1650s ^[1]. Christopher Wren — the same man who would rebuild London after the 1666 fire — rectified the cycloid in 1658, proving each arch is exactly 8a, eight times the rolling circle's radius, a clean whole number that startled everyone ^[1]^[2]. But those were transcendental curves. The prize was an algebraic one.

That fell to William Neile, a young Oxford student, who in 1657 found the arc length of the semicubical parabola y² = x³ — the first algebraic curve besides the line and circle ever rectified ^[3]^[4]. His teacher John Wallis published the method in his De Cycloide in 1659, crediting Neile ^[3]^[4]. Almost simultaneously and independently, Hendrik van Heuraet in the Netherlands and Pierre de Fermat in France worked out the general trick: turn a length problem into an area problem ^[4]^[5]. Van Heuraet published in 1659 ^[4]; Fermat followed in 1660 with a geometric dissertation stating the same result ^[1]. Between them they'd built the integrand √(1 + (dy/dx)²) in everything but modern notation — a full generation before Newton and Leibniz packaged calculus. When the ellipse refused to yield to any of their methods, the frustration eventually grew into the theory of elliptic integrals ^[6].

Related tools

/tools/belt-pulley-speed
/tools/convert-length
/tools/convert-angle

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →