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
| Field | What to enter |
|---|---|
| Type of curve | Choose one: Unweighted catenary y = a·cosh(x/a), Weighted catenary y = b·cosh(x/a) |
| Parameter a | A number |
| Parameter b | A number |
| x | A number |
Example values
How the main result changes when Parameter a changes and everything else stays at the example values above:
| Parameter a | y(x) |
|---|---|
| 5 | 12.8873 |
| 7.5 | 12.1869 |
| 10 | 13.3743 |
| 12.5 | 15.1486 |
| 15 | 17.1844 |
| 20 | 21.6214 |
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
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.
Frequently asked questions
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.