Catenary Curve Calculator - Hanging Chain and Cable Sag
Calculate sag height, slope, arc length, and tension for catenary curves using the hyperbolic cosine formula.
Enter the catenary parameter a and horizontal position x to compute curve properties for hanging chains and cables.
Catenary Curve Calculator - Hanging Chain and Cable Sag
Calculate sag height, slope, arc length, and tension for catenary curves using the hyperbolic cosine formula.
Vertical sag y = a·cosh(x/a) − a at horizontal position x from the lowest point.
Must be positive — equals T₀/w (horizontal tension divided by linear weight density)
About the Catenary Curve Calculator
A catenary is the curve formed naturally by a flexible, uniform chain or cable hanging freely under its own weight between two fixed endpoints. The word comes from the Latin catena, meaning chain. Despite looking like a parabola, the catenary follows the hyperbolic cosine function, a fundamentally different mathematical shape.
The equation of a catenary with its lowest point at the origin is y = a·cosh(x/a) − a, where the parameter a = T₀/w. Here T₀ is the horizontal component of the tension (equal everywhere along the cable), and w is the weight per unit length of the cable. A larger value of a means the cable is either lighter or under more tension, resulting in a flatter sag curve.
The slope at any point is dy/dx = sinh(x/a). The arc length along the curve from the lowest point (x = 0) to horizontal position x is s = a·sinh(x/a). The total tension at any point along the cable is T = w·a·cosh(x/a), meaning tension is greatest at the endpoints and smallest at the lowest point.
Catenary analysis is essential in civil engineering for suspension bridges, overhead power lines, and cable-supported roof structures. The Golden Gate Bridge, for example, has main cables that follow nearly catenary shapes under their own self-weight, though the addition of the road deck load shifts them closer to a parabola. Overhead electric power lines also sag in catenary shapes between towers, and engineers must calculate this sag to ensure minimum ground clearance.
In architecture, the inverted catenary (catenary arch) is the ideal shape for a freestanding arch under its own weight, since every section is under pure compression with no bending stress. The Gateway Arch in St. Louis is a flattened catenary arch, and the Sagrada Família's vaulting was designed using hanging chain models.
The hyperbolic functions cosh and sinh appear naturally in catenary analysis and are available in all scientific calculators and programming languages. This calculator uses them directly, so you can compute sag heights, slopes, arc lengths, and tension values for any combination of parameter a and position x without manual integration.
Catenary Curve Examples
Typical catenary calculations for engineering and physics applications.
| Parameters | Result | Calculation |
|---|---|---|
| a = 10, x = 5 (sag) | 1.2760 | y = 10·cosh(0.5) − 10 ≈ 1.276 m sag at half-span |
| a = 10, x = 10 (arc length) | 11.7520 | s = 10·sinh(1) ≈ 11.752 m cable length from center to end |
| a = 50, x = 30 (slope) | 0.6366 | dy/dx = sinh(0.6) ≈ 0.637 — about 32.5° angle |
| a = 100, x = 0 (sag) | 0 | At the lowest point (x = 0) sag is always zero |
| a = 20, x = 15 (tension) | 25.8937 | T/w = 20·cosh(0.75) ≈ 25.89 — normalized tension factor |
How to Use the Catenary Curve Calculator
- Select the calculation type: Sag Height, Slope, Arc Length, or Tension Factor.
- Enter the catenary parameter a in the Parameter a field. This equals T₀/w (horizontal tension divided by linear weight density). It must be a positive number.
- Enter the horizontal position x in the Position x field. Use x = 0 for the lowest point; use half the span for endpoint values.
- Click Calculate to see the result along with the formula used.
- Click Reset to clear both fields and start a new calculation.
Catenary Curve FAQ
What is a catenary curve?
A catenary is the shape formed by a flexible, uniform chain or cable hanging freely under its own weight between two fixed points. Its equation is y = a·cosh(x/a) − a, where a = T₀/w is the catenary parameter (horizontal tension divided by linear weight density). Despite resembling a parabola, the catenary is a distinctly different and more accurate model for hanging cables.
How is catenary different from a parabola?
A parabola describes the shape of a cable when the load is distributed uniformly along the horizontal span (as in a suspension bridge with a very heavy road deck). A catenary describes the shape when the load is the cable's own weight distributed along its length. For shallow sags the two curves look nearly identical, but they diverge significantly for steeper chains and cables.
What is the catenary parameter a?
The parameter a equals T₀/w, where T₀ is the horizontal component of the cable tension (constant everywhere) and w is the weight per unit length of the cable. A larger a means the cable is tighter or lighter, resulting in a flatter shape. The sag at any point is y = a·cosh(x/a) − a.
How do I find the arc length of a catenary?
The arc length from the lowest point (x = 0) to horizontal position x is s = a·sinh(x/a). This gives the actual cable or chain length along the curve, which is always longer than the horizontal distance. For the full span from −L to +L, the total cable length is 2a·sinh(L/a).
What are real-world applications of catenary calculations?
Catenaries appear in suspension bridge cable design, overhead electric power line sag analysis, cable-supported roof structures, and chain drives. The inverted catenary is the ideal arch shape for structures under pure compression, used in the Gateway Arch and Sagrada Família. Engineers use catenary calculations to ensure minimum ground clearances and to calculate material lengths.
Why must parameter a be positive?
The parameter a represents a physical ratio of tension to weight per unit length, both of which are positive quantities. A negative or zero value of a has no physical meaning for a hanging cable. Mathematically, a = 0 would make the catenary equation undefined due to division by zero, so the calculator requires a > 0.