Catenary Curve Calculator

Compute the height, arc length, slope angle, and tension ratio of a catenary curve — the shape a chain or cable forms when hanging freely between two supports — using y = a·cosh(x/a).

Quick Facts

Formula
y = a·cosh(x/a)
The catenary equation for a chain or cable hanging freely under its own weight, symmetric about its lowest point (the vertex).
Parameter a
a = H / w
Horizontal tension H divided by weight per unit length w. A larger a gives a flatter curve; a smaller a gives a tighter sag.

Your Results

Calculated
Curve height (y)
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Height at x on the curve
Arc length from vertex
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Chain/cable length from the lowest point to x
Slope angle
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Angle of the tangent line at x
Tension ratio (T / H)
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Local tension relative to horizontal tension

Ready

Set the catenary parameter a and a horizontal distance x, then press Calculate.

How the Catenary Curve works

A catenary is the exact curve a flexible chain, rope, or cable traces when it hangs freely between two supports and sags under its own weight. Unlike a parabola — the shape used to approximate suspension-bridge cables carrying a level roadway — a true hanging chain follows the hyperbolic cosine function. This calculator evaluates that function directly from the inputs you provide, so you can find the curve's height, arc length, and local slope at any horizontal position.

Formula and method

The catenary equation is y = a·cosh((x − h)/a) + k, where a is the catenary parameter, x is the horizontal position, and h and k shift the vertex (the curve's lowest point) horizontally and vertically. The parameter a equals the horizontal tension H in the chain divided by its weight per unit length w (a = H/w): a large a means high tension relative to weight, giving a flat, gently curving line; a small a means the curve sags sharply. From this base formula, three related quantities follow directly:

  • Slope: the tangent's slope at x is dy/dx = sinh((x − h)/a); the slope angle is atan of that value.
  • Arc length: the length of curve from the vertex to x is s = a·sinh((x − h)/a).
  • Tension ratio: the tension T at any point relates to the horizontal tension H by T = H·cosh((x − h)/a), so T/H = cosh((x − h)/a) — the same factor that scales the height above the vertex.

Common sources of error

  • Confusing a catenary with a parabola: the two curves look alike for small sags but diverge for steep ones — always use cosh/sinh, not a squared term, for a hanging chain or cable.
  • Using a non-positive parameter: a must be greater than 0; it represents a physical ratio of tension to weight and cannot be zero or negative.
  • Rounding early: cosh and sinh grow quickly, so carry full precision through intermediate steps rather than rounding a or x before computing y, s, or the slope.

Checking your result

At the vertex (x = h) the height equals a + k, the slope is 0, and the tension ratio is exactly 1 — use these as a quick sanity check. Moving away from the vertex in either direction, height, arc length, and tension ratio should all increase, and the curve should be symmetric: two points equally spaced from h on either side give the same height, arc length, and tension ratio (with opposite-signed slope).

Applications

The catenary shows up anywhere a flexible line hangs under gravity alone: suspended power lines and telephone wire, ship anchor chains, the cables of some arch bridges, and architectural arches designed as an inverted catenary for pure compression (as used by Antoni Gaudí and in the Gateway Arch's design). Knowing a lets you predict clearance, required cable length, and how tension changes along the span.

Frequently Asked Questions

What is a catenary curve?
A catenary is the curve formed by a flexible chain or cable hanging freely between two supports under its own weight. It is described by the equation y = a·cosh(x/a), where a is the catenary parameter and x is measured horizontally from the curve's lowest point (the vertex).
What does the parameter a represent?
The parameter a equals the horizontal tension H divided by the weight per unit length w of the chain or cable (a = H/w). A larger a produces a flatter, more stretched-out curve, while a smaller a produces a tighter, more sharply curved sag.
Is a catenary the same shape as a parabola?
No, although the two can look similar for shallow sags. A catenary follows the hyperbolic cosine function y = a·cosh(x/a), while a parabola follows a quadratic function. Galileo is sometimes credited with an early guess that a hanging chain traces a parabola, but in 1691 Leibniz, Huygens, and Johann Bernoulli independently showed the true curve is the hyperbolic cosine catenary.
How do I find the arc length along a catenary?
The arc length from the vertex to a point at horizontal distance x is s = a·sinh(x/a). This gives the length of chain or cable needed between the lowest point and that point on the curve.

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