Calculate catenary curve properties including sag, arc length, tension, slope, and curvature using the equation y = a·cosh(x/a).
The catenary curve describes the shape formed by a flexible chain or cable hanging freely under its own weight, supported at its endpoints. Defined by the equation y = a·cosh(x/a), it is one of the most elegant curves in mathematics and engineering, appearing everywhere from power line design to architectural arches.
The parameter "a" in the catenary equation represents the ratio of horizontal tension to the weight per unit length of the cable. A larger value of "a" produces a flatter, more gently curving catenary, while smaller values create deeper, more pronounced sags. Understanding this relationship is critical for structural engineers designing suspension bridges and electrical engineers planning overhead transmission lines.
This calculator computes all essential catenary properties: the height at any point along the curve, total sag between supports, arc length of the cable, horizontal and maximum tensions, local slope, curvature, and radius of curvature. It also generates a detailed properties table showing how these values change along the curve's length.
Preset configurations for common real-world scenarios — power lines, suspension bridges, heavy chains, decorative cables, and clotheslines — let you quickly explore how different parameters affect the curve's shape. The interactive curve profile visualization shows the catenary's characteristic U-shape, and the reference table summarizes all key formulas.
Whether you're an engineer sizing cables, a physics student studying statics, or an architect designing hanging structures, this tool provides the precise catenary calculations you need.
Catenary equations involve the hyperbolic cosine function with parameters that interact nonlinearly — computing y-coordinates, arc lengths, sag depths, and horizontal tension from a cable's weight per unit length and support span requires careful algebra that is easy to get wrong. This calculator evaluates the full catenary profile instantly, plotting the curve for any parameter a, computing sag, arc length, and tension, and letting you switch between metric and imperial units. Architects, structural engineers, and power-line designers use catenary math daily, and this tool removes the error-prone manual steps so you can focus on design decisions.
y = a · cosh(x / a); Arc Length s = a · sinh(x / a); Sag = a · cosh(L / 2a) − a; T_horizontal = w · a; T_max = w · a · cosh(L / 2a)
Result: y ≈ 54.03, sag ≈ 4.03 m, arc length ≈ 40.43 m
With a = 50 m and span = 40 m, the catenary sags about 4 m below the support level. The total cable length is slightly greater than the horizontal span.
A common misconception is that a hanging cable forms a parabola. Galileo originally believed this, but in 1691 Leibniz, Huygens, and Johann Bernoulli independently proved the true shape is a catenary: y = a cosh(x/a). A parabola is a good approximation only when the sag is small relative to the span. For deeper sags, the hyperbolic cosine diverges significantly from a parabolic fit. The parameter a = T₀/w represents the ratio of horizontal tension to weight per unit length and controls the curve's "flatness" — larger a means a flatter, tighter cable.
The inverted catenary is the ideal shape for a compression arch: the Gateway Arch in St. Louis, Gaudí's hanging-chain models for the Sagrada Família, and Roman aqueduct arches all exploit this principle. Power transmission lines, suspension bridge cables (under uniform horizontal load, these are parabolas, but under self-weight, catenaries), and overhead contact wires for electric railways all follow catenary geometry. Understanding sag is critical: excessive sag causes ground clearance violations, while too little sag (high tension) risks conductor fatigue or snap in cold weather.
The arc length of a catenary from x = −L/2 to L/2 is 2a sinh(L/(2a)), which is always longer than the span L. Engineers must order enough cable to cover this extra length. The tension varies along the curve: minimum at the lowest point (T₀ = wa) and maximum at the supports T = w√(a² + y²). These relationships drive conductor selection, tower height decisions, and span-to-sag ratio tables in power-line design. Wind and ice loading add uniform horizontal and vertical loads that shift the curve between a catenary and a parabola, requiring iterative sag-tension calculations.
A catenary is the curve formed by a uniform flexible chain or cable hanging freely under gravity, described by y = a·cosh(x/a). Use this as a practical reminder before finalizing the result.
A parabola results from uniform horizontal loading (like a suspension bridge deck), while a catenary results from uniform loading along the cable itself. They are nearly identical for small sag-to-span ratios.
Parameter a equals horizontal tension divided by weight per unit length (T_h / w). Larger a means a flatter curve.
Sag = a · cosh(L / 2a) − a, where L is the span. This gives the vertical distance from the support level to the lowest point.
Maximum tension occurs at the support points, calculated as T_max = w · a · cosh(L / 2a). Keep this note short and outcome-focused for reuse.
Power lines between poles, anchor chains, spider webs, the Gateway Arch in St. Louis (an inverted catenary), and freely hanging necklaces all form catenary shapes.
This calculator assumes level supports. Inclined catenaries require asymmetric formulations with different support heights.