Catenary Curve Calculator — Hanging Cable Length & Shape

Formula

How this is calculated

A catenary (from the Latin catena, chain) is the shape taken by a uniform, inextensible, flexible cable hanging between two points of equal height under its own weight and gravity alone. The shape is described by the hyperbolic cosine: y = a · cosh(x/a), where a is the catenary constant (the height of the lowest point above a reference level, or equivalently the ratio of horizontal tension to cable weight per unit length).

Given span 2L (total horizontal distance) and sag d (vertical drop from support to lowest point), the catenary constant a is found by solving the transcendental equation a·cosh(L/a) − a − d = 0 numerically, using Newton's method with the small-sag starting guess a₀ = L²/(2d). Once a is known, the arc length is S = 2a·sinh(L/a), which is always greater than the straight-line span. The arc-to-span ratio reveals how much extra cable is needed compared to the straight-line distance.

Assumptions and limitations: the cable is uniform (constant mass per unit length), perfectly flexible (no bending stiffness), and inextensible (does not stretch). Real cables stretch under load (add elastic elongation separately). Stiff wire ropes, beams, or chains with significant bending stiffness may deviate from the pure catenary. Power-line sag calculations usually account for wind load, ice loading, and temperature-dependent elastic stretch on top of the catenary geometry.