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Catenary Calculator

Calculate the catenary curve — the height y = a·cosh(x/a) of a hanging chain at any position x, step by step with the formula, worked examples and FAQ.

Quick answer
How do you calculate the catenary curve?
A freely hanging chain follows the curve y = a·cosh(x/a). The parameter a sets the shape and is also the height of the lowest point (at x = 0, y = a). Example a = 2, x = 3: y = 2·cosh(1.5) ≈ 4.70.
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Step-by-step
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HowTo

Catenary curve — 3 steps

Using a = 2, x = 3
  1. 1
    Step 1 of 3

    Set the parameter and position

    Choose the parameter a (> 0) and the position x where you want the height, e.g. a = 2 and x = 3.

  2. 2
    Step 2 of 3

    Divide x by a

    Form the ratio x/a = 3/2 = 1.5 — the argument of the hyperbolic cosine.

  3. 3
    Step 3 of 3

    Compute with a·cosh

    y = a·cosh(x/a) = 2·cosh(1.5) ≈ 2·2.3524 ≈ 4.70.

Examples

Catenary curve — worked examples

Height y at various positions
a=2, x=3
2·cosh(3/2)
2·cosh(1.5)
≈ 4.70
a=2, x=0
2·cosh(0)
2·1
= 2 (lowest point)
a=1, x=1
1·cosh(1)
≈ 1.54
a=5, x=5
5·cosh(1)
≈ 7.72
a=3, x=6
3·cosh(2)
≈ 11.29
Theory

The catenary and the hyperbolic cosine

When a flexible chain of uniform weight hangs from just its two ends, it settles under its own weight into the shape of the catenary: y = a·cosh(x/a), where cosh is the hyperbolic cosine, cosh(t) = (eᵗ + e⁻ᵗ)/2. The parameter a controls how flat or steep the curve runs and equals the height of the lowest point above the reference line: at x = 0, cosh(0) = 1, so y = a. As |x| grows, y rises slowly at first, then ever more steeply. The catenary looks similar to a parabola but is not the same curve — it falls away a little more gently. Turned upside down it gives the ideal shape of a self-supporting arch, which is why it appears in architecture (such as the Gateway Arch) and bridge design. Hyperbolic functions like cosh belong to upper-secondary calculus.

Pitfalls

Common mistakes

Confusing cosh with cos

The catenary uses the hyperbolic cosine cosh, not the ordinary cosine cos. cosh(0) = 1, but cosh grows without bound.

Using a = 0 or negative

The parameter a must be greater than 0, otherwise x/a is not meaningfully defined.

Forgetting the factor a

It is a·cosh(x/a), not just cosh(x/a). The leading a scales the height.

Treating it as a parabola

y = a·cosh(x/a) is not a parabola y = c·x²; they only resemble each other near the lowest point.
FAQ

Frequently asked questions

Glossary

Glossary — key terms explained simply

Catenary
Curve of a freely hanging chain, y = a·cosh(x/a).
Parameter a
Shape factor and height of the lowest point.
cosh
Hyperbolic cosine, (eᵗ + e⁻ᵗ)/2.
Lowest point
Position x = 0, where y = a.
Rise
Height above the lowest point, y − a.
Arch
An inverted catenary as a self-supporting shape.