# Catenary Curve Calculator

URL: https://speedycalculator.com/calculators/math/catenary-curve/
Last updated: 2026-10-30

> The catenary calculator evaluates the hanging-chain curve y = a·cosh(x/a) (or a weighted form y = b·cosh(x/a)) at any x, with the arc length, slope and sag for a span from −x to x.

## Inputs

- Type of curve
- Parameter a
- Parameter b
- x

## Results

- y(x)
- Arc length from 0 to x
- Slope dy/dx
- Sag over the span −x…x
- Lowest point y(0)

## Formula

```
y = a·cosh(x/a),  arc length from 0 to x: s = a·sinh(x/a)
Sag over the span −x…x: a·cosh(x/a) − a
```

## Explanation

A chain or cable hanging under its own weight forms a catenary, not a parabola. The shape (inverted) gives the strongest arches – Gaudí used it, and the Gateway Arch in St. Louis is a weighted catenary.

### How to use the Catenary Curve Calculator

- **Type of curve**
- **Parameter a**
- **Parameter b**
- **x**

Results update as you type, and everything is calculated in your browser.

### How it works

The parameter a is the ratio of horizontal tension to weight per unit length; larger a gives a flatter curve. The weighted form scales the height by b while keeping the horizontal scale a.

### Inputs explained

FieldWhat to enterType of curveChoose one: Unweighted catenary y = a·cosh(x/a), Weighted catenary y = b·cosh(x/a)Parameter aA numberParameter bA numberxA number

### Example values

How the main result changes when **Parameter a** changes and everything else stays at the example values above:

Parameter ay(x)512.88737.512.18691013.374312.515.14861517.18442021.6214

## Worked example

Inputs — Type of curve: Unweighted catenary y = a·cosh(x/a); Parameter a: 10; x: 8.

Results — **y(x) = 13.37434946**; Arc length from 0 to x = 8.88105982; Slope dy/dx = 0.88810598; Sag over the span −x…x = 3.37434946; Lowest point y(0) = 10.

## Key terms

- **Catenary**: The curve y = a cosh(x/a) of a hanging chain.

## FAQ

**Is a hanging chain a parabola?**
No – it is a catenary; a parabola is the shape of a cable carrying a uniform horizontal load, such as a suspension bridge deck.

**What does a represent?**
Horizontal tension divided by weight per unit length.

**Why are catenary arches strong?**
An inverted catenary is in pure compression, with no bending.

**How do I find a from a measured sag?**
Solve a·cosh(L/2a) − a = sag numerically for half-span L/2.


## Sources

- [Wolfram MathWorld – Catenary](https://mathworld.wolfram.com/Catenary.html)
- [OpenStax – Calculus Volume 1](https://openstax.org/details/books/calculus-volume-1)