Catenary Curve Calculator
A catenary is the curve a perfectly flexible, uniform cable makes when it hangs freely between two supports of equal height. Enter the horizontal span and either the midpoint sag or the shape parameter a, and the calculator finds the arc length (true cable length), catenary parameter, horizontal tension, maximum tension at the supports, and the cable angle at each support. Results update instantly as you type, and a live chart shows the curve cross-section.
Formula
Worked example
A cable spans 20 m with a midpoint sag of 2 m. Newton-Raphson gives a = 25.67 m. Arc length S = 2 * 25.67 * sinh(20 / 51.34) = 20.262 m, about 1.3% longer than the span. The cable leaves each support at atan(sinh(0.389)) = 21.4 degrees. If the cable weighs 1 N/m, horizontal tension H = 25.67 N and maximum tension at the supports T = 25.67 * cosh(0.389) = 28.2 N.
What is a catenary curve?
A catenary is the shape a uniform, flexible, inextensible cable forms when it hangs freely between two points of equal height under its own weight. The word comes from the Latin "catena" (chain), and the shape was first described mathematically by Leibniz, Huygens, and Johann Bernoulli in 1691, following a challenge posed by Jakob Bernoulli. Despite looking like a parabola, the catenary is a distinct curve described by the hyperbolic cosine: y = a * cosh(x / a), where the parameter a is the ratio of the horizontal tension to the cable weight per unit length. When a is large relative to the span, the cable is nearly taut and the curve is very flat. When a is small, the cable hangs deeply. The catenary appears throughout engineering and architecture: overhead power lines, suspension bridge cables, arch bridges (an inverted catenary is the ideal compression arch), and the sail of a wind-blown sailboat all approximate catenary shapes.
How to use this calculator
Choose metric or imperial units, then pick a solve mode. In "given span and sag" mode, enter the horizontal distance between the supports and how far the cable sags below the support level at its lowest point; the calculator solves for the catenary parameter and all other quantities. In "given span and parameter a" mode, enter the span and the value of a directly; the calculator derives the sag. In both modes, optionally enter the cable weight per unit length (N/m in metric, lb/ft in imperial) to compute the horizontal tension and the peak tension at the supports. The chart at the bottom draws the cable cross-section to scale, and the steps panel shows every calculation with your actual numbers substituted in.
The catenary formulas and their meaning
The catenary rests on two fundamental relationships. The sag formula d = a * (cosh(L / (2a)) - 1) tells you the vertical drop at the midpoint given the span L and parameter a. The arc length formula S = 2 * a * sinh(L / (2a)) gives the true cable length needed to achieve that span and sag; because sinh grows faster than the span, S is always greater than L. The angle at which the cable meets each support is theta = atan(sinh(L / (2a))), which is also the angle the support anchor must resist. If the cable has a known weight per unit length w, then a = H / w, where H is the horizontal component of tension - constant everywhere along the cable by equilibrium. The maximum tension occurs at the supports: T_max = H * cosh(L / (2a)), which equals the horizontal tension amplified by the curvature at the endpoints.
Engineering applications: power lines, bridges, and arches
Power lines are designed using catenary analysis because the true cable length, sag clearance to the ground, and conductor tension must all be within specification over a range of temperatures (thermal expansion changes a). Suspension bridge designers use an approximate catenary (parabola for a uniformly loaded deck) for the main cables, then work backwards to size the hangers and deck. Inverted, the catenary becomes the most efficient compression arch: a structure loaded uniformly along its horizontal projection and shaped as a catenary has no bending moment anywhere, only pure compression. The Gateway Arch in St. Louis and Gaudi's Sagrada Familia columns follow this principle. Mooring chains for ships and oil platforms also rely on catenary theory: the sag in the chain absorbs shock loads from waves, protecting the anchor.
Typical sag-to-span ratios by application
| Application | Typical sag/span | Notes |
|---|---|---|
| High-voltage transmission line | 1-3% | Low sag reduces right-of-way clearance needs |
| Distribution power line | 3-6% | Balances tension, conductor weight, and pole spacing |
| Overhead contact wire (rail) | 0.5-1% | Must remain nearly flat for pantograph contact |
| Suspension bridge main cable | 8-12% | Stiffened deck reduces dynamic wind response |
| Footbridge or cable walkway | 5-15% | Higher sag lowers cable tension and anchor forces |
| Decorative string lights | 10-25% | Aesthetics drive the choice; structural loads are minimal |
| Ship mooring line (catenary effect) | 5-20% | Sag absorbs shock loads from wave motion |
| Ski lift haul rope | 1-4% | Tight to minimize vertical displacement of carriers |
Industry design guidelines for sag as a percentage of horizontal span.
Frequently asked questions
What is the catenary parameter a?
The parameter a is the characteristic length of the catenary, defined as the ratio of horizontal tension H to the cable weight per unit length w: a = H / w. When a is large the curve is nearly flat (high tension relative to weight); when a is small the cable hangs in a deep arc. Geometrically, a is also the height of the lowest point of the catenary above the directrix (the horizontal line y = 0 in the canonical form). If you know the span and sag, this calculator solves for a numerically.
Is a catenary the same as a parabola?
No. A catenary describes a cable loaded by its own weight per unit arc length. A parabola describes a cable loaded uniformly per unit horizontal distance, which is approximately true for the main cables of a stiffened suspension bridge (where the deck load dominates). For sag-to-span ratios below about 5%, the two curves differ by less than 0.1% and the parabolic approximation is often used in practice. For deeper sags, only the catenary formula is accurate.
Why is the cable longer than the horizontal span?
Because the cable follows a curved path and any curve is longer than its chord (the straight line between the endpoints). The arc length formula S = 2a * sinh(L / (2a)) always returns a value greater than L. For a shallow catenary (small sag) the extra length is tiny: a 2% sag gives roughly a 0.5% longer cable. For a 15% sag the cable is about 3.7% longer. Knowing the exact extra length is important when ordering cable, designing splice locations, or calculating thermal expansion.
What happens to tension as sag increases?
For a given span, more sag means lower tension: a shallower catenary (small sag) requires high tension to hold the cable nearly straight, while a deeply sagging cable is under much less tension. The relationship is inverse and nonlinear. Doubling the sag roughly halves the horizontal tension. This trade-off drives power line design: engineers accept more sag to reduce the tension that poles and anchors must resist, but they must maintain minimum ground clearance.
How do I find the cable length needed for a given span and sag?
Use the arc length formula S = 2a * sinh(L / (2a)), where a is found first by solving a * cosh(L / (2a)) - a = d (the sag equation) numerically. This calculator does both steps automatically. Enter the span L and the desired midpoint sag d in "given span and sag" mode, and the arc length field shows the required cable length.
Can I use this calculator for unequal support heights?
This calculator assumes both supports are at the same height (a symmetric catenary). For supports at different heights, the catenary equation still applies, but the lowest point is no longer at the midspan and a different set of equations must be solved. A common approach splits the asymmetric catenary into two symmetric half-catenaries and solves for the parameter a iteratively.
Why is the catenary the ideal arch shape?
An inverted catenary carries loads entirely in compression with no bending, which lets masonry arches use materials (stone, brick, concrete) that are strong in compression but weak in tension. Gaudi discovered this by hanging chains to find arch shapes and then inverting the chain model. A catenary arch under uniform self-weight loading has zero bending moment at every cross-section, making it structurally perfect for gravity loads.