Hang a uniform flexible chain from two points and it settles into a curve that has a precise equation: y = a cosh(x / a), the hyperbolic cosine. That curve is the catenary, from the Latin for chain. The catenary curve calculator above finds the parameter a from whichever pair of quantities you have, then reports the sag, the exact arc length, the ratio of tension at the ends to tension at the bottom, the slope where the curve meets its supports, and how far the familiar parabolic approximation is out for your particular numbers.
Arb Digital publishes this as a mathematics and teaching tool. It is the curve itself. Its engineering sibling, our cable sag calculator, works the same physical situation the way overhead line and rigging practice does — from span, weight per unit length and tension, using the parabolic approximation the trades work in, and with the full set of structural cautions attached. This page is the exact mathematics behind that approximation. Neither is a substitute for a design: a real cable is sized, specified and inspected by a qualified engineer, and nothing here states any capacity.
What This Catenary Curve Calculator Does
Everything about a symmetric catenary follows from two numbers: the span and the parameter a. The difficulty is that you almost never have a. You have the span and the sag, or the span and the length of chain, and a has to be extracted from an equation it cannot be isolated in. The tool solves that numerically and then reports every other property in closed form.
It also runs the comparison that makes the page worth having. For your span and sag it computes the exact catenary arc length and the parabolic estimate side by side, and reports the difference as a percentage. That single output settles the recurring question of when the approximation is good enough, without anyone having to trust a rule of thumb.
The optional weight per unit length converts the dimensionless results into forces. That conversion is trivial in form and profound in meaning: the horizontal tension is exactly the weight per unit length times a. The parameter is not an abstract fitting constant — it is the tension divided by the weight, expressed as a length.
How to Use It
- Pick the pair you actually measured. Span and sag is the usual case; span and chain length is the one you get when the material is already cut.
- Check that the length exceeds the span. In length mode a curve shorter than the straight line between its ends does not exist, and the tool says so.
- Read a as a physical quantity. A large a means a flat, tightly pulled curve; a small a means a deep, slack one.
- Look at the parabola comparison. It tells you directly whether the simpler engineering form is adequate for the sag ratio you have.
- Use the tension ratio, not just the sag. The ends always carry more tension than the bottom, and the ratio is the cosine hyperbolic of half the span over a.
The Formula / How It's Calculated
Place the origin at the lowest point of the curve so that the equation is y = a cosh(x / a), and put the two ends at x = ±L/2, both at the same height. The definitions of the hyperbolic functions used throughout are set out in section 4.28 of the NIST Digital Library of Mathematical Functions, on definitions and periodicity of hyperbolic functions.
The sag between the ends and the lowest point is d = a (cosh(L / 2a) − 1). The exact arc length is S = 2a sinh(L / 2a). The slope at the end is dy/dx = sinh(L / 2a), so the angle from horizontal there is arctan(sinh(L / 2a)).
The mechanics falls out of the same parameter. With weight w per unit length, horizontal tension is H = w a everywhere along the curve, and the total tension at any point is T = w y, proportional to the height above the directrix. At the ends that gives T = w (a + d), so the ratio of end tension to lowest tension is simply cosh(L / 2a). As Wolfram MathWorld's Catenary entry records, Jungius disproved Galileo's claim that the curve was a parabola in 1669, and Leibniz, Huygens and Johann Bernoulli each derived the true equation independently in 1691.
Neither the sag equation nor the length equation can be rearranged for a in elementary terms, so the tool brackets the solution and bisects it. Sag increases monotonically as a decreases, which makes the search well behaved and convergence certain.
Worked example with the loaded values. For a span of 100 and a sag of 10, the solve returns a = 126.632. The exact arc length is 2 × 126.632 × sinh(50 / 126.632) = 102.6187, while the parabolic estimate L + 8d²/3L gives 102.6667 — high by 0.047 per cent. The end-to-bottom tension ratio is cosh(0.39485) = 1.0790, and the ends meet their supports at arctan(sinh 0.39485) = 22.06 degrees from horizontal. At 10 per unit length the horizontal tension is 1,266.3 and the end tension is 1,366.3.
Catenary Against Parabola
The two curves are genuinely different, and the difference has a clean physical explanation. A catenary arises when the load is uniform along the curve — a chain, where every unit of arc weighs the same. A parabola arises when the load is uniform along the horizontal — a suspension bridge, where the deck below is what weighs something and the cable's own mass is comparatively small. The main cables of a suspension bridge really are close to parabolic, and a bare hanging cable really is a catenary.
Where they agree is at small sag. Expand the hyperbolic cosine as a series and the first two terms give exactly the parabola; everything beyond that is the correction. So the parabolic error grows with the fourth power of the sag ratio and is utterly negligible in the sag range most tensioned cables operate in, which is why generations of engineers have used it without difficulty. The worked example above shows the error at a sag of a tenth of the span running under one twentieth of one per cent.
At large sag ratios that changes. A rope hanging slack, a chain barrier between posts, a decorative festoon, a mooring line — these can sag by a third of the span or more, and there the parabola misstates both the length and the tension by amounts that matter. The rule that emerges is simple: use the parabola for taut spans, use the catenary for slack ones, and if you are unsure, run both and look at the difference, which is exactly what this page prints for you.
Where the Curve Turns Up
The inverted catenary is the shape a masonry arch takes when it carries only its own weight in pure compression, which is why the form recurs in structures with no tensile capacity to spare. The Gateway Arch in St Louis is a well-known approximation to an inverted catenary, though its cross-section varies along its height so it is not a pure one. Our arch calculator works with circular arches, which are what most building drawings actually specify, and the boundary between the two pages is exactly that: circular geometry for setting out an opening, hyperbolic geometry for the funicular shape.
The curve also appears in the roulette that generates it — a catenary is traced by the focus of a parabola rolling along a straight line — and in the surface of revolution it produces, the catenoid, which is the minimal surface formed by a soap film between two rings. That is not a coincidence: both the hanging chain and the soap film minimise a quantity, potential energy in one case and surface area in the other.
And it appears wherever something flexible hangs: overhead line conductors, cableways, tent ridges, suspended lighting, mooring chains, festoon cabling and the anchor line of a moored vessel, where the shape of the catenary on the seabed is what provides the compliance in the mooring. For the underlying arc-length mathematics in general, our arc length calculator covers the circular case and the trigonometric functions calculator the circular functions that the hyperbolic ones parallel.
Reading the Parameter a
It is worth spending a moment on what a means, because it is the least intuitive part of the subject and the most useful once it clicks. Geometrically, a is the height of the lowest point of the curve above the directrix — a horizontal line below the curve that the equation is measured from. Physically, a is the horizontal tension divided by the weight per unit length. Those are the same number.
That gives it a scale-free character. A curve with a span of 100 and a parameter of 126.6 has exactly the same shape as one with a span of 1 and a parameter of 1.266; only the size differs. What controls the shape is the ratio L/a, and everything this page reports — the sag ratio, the tension ratio, the end angle, the parabolic error — depends on that ratio alone.
It also explains something that surprises people. A heavy chain and a light string, pulled to the same horizontal tension, do not hang in the same curve: the heavier one has a smaller a and sags more. But a heavy chain and a light string at the same ratio of tension to weight hang identically. Weight alone does not determine the shape; the ratio does.
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Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Calling a hanging chain a parabola — it is a hyperbolic cosine, and the two only coincide in the small-sag limit.
- Entering a curve length shorter than the span — no curve can be shorter than the straight line between its own endpoints.
- Treating a as a fitting constant — it is a length with a meaning: horizontal tension divided by weight per unit length.
- Assuming the tension is the same everywhere — only the horizontal component is constant. Total tension rises with height, so the ends always carry more than the bottom.
- Applying the symmetric equations to unequal supports — everything here assumes both ends are at the same height, and an inclined span is a different problem.
Related Free Tools From Arb Digital
For the engineering sag-and-tension case in the form the trades use, see the cable sag calculator. For circular arch geometry, the arch calculator, and for circular arcs in the abstract the arc length calculator and circle calculator. The trigonometric functions calculator covers the circular functions, and the beam load calculator the structural side. Everything we publish is listed on the free online tools hub.
Frequently Asked Questions
With the origin at the lowest point, y = a cosh(x / a), where cosh is the hyperbolic cosine and a is the catenary parameter. It is the curve a uniform flexible chain takes when hung from two points under gravity.
No. Galileo believed so, and Jungius disproved it in 1669. A chain, whose weight is uniform along its own length, forms a catenary. A parabola arises when the load is uniform along the horizontal instead, as with a suspension bridge deck.
Geometrically it is the height of the lowest point above the directrix. Physically it is the horizontal tension divided by the weight per unit length. A larger a means a flatter, more tightly pulled curve.
Because the parameter cannot be isolated algebraically from either the sag equation or the length equation. Both put a inside and outside a hyperbolic function, so the solution is found by bisection instead.
Very close at small sag, because the parabola is the first two terms of the series expansion of the catenary. At a sag of a tenth of the span the arc length error is under a twentieth of one per cent. It grows quickly on slack curves.
At the supports. The horizontal component is constant along the whole curve, but total tension is proportional to height above the directrix, so it is smallest at the lowest point and largest at the ends.
An inverted catenary is the shape that carries only its own weight in pure compression, which is why it recurs in masonry. Most building arches are specified as circular arcs instead, which is a different geometry.
No. This page is mathematics. Sag and tension on a real cable also change with temperature, creep, ice and wind, and any load-bearing cable must be designed, specified and inspected by a qualified engineer.
This tool solves the mathematics of the catenary from values you supply, for education and analysis only. It is not an engineering design, it states no capacity of any kind, and any load-bearing cable or structure must be designed and inspected by a qualified engineer under the applicable standard.
