Hong Life CalculatorsUnderstanding the catenary curve calculator — concept and principle
This calculator finds the shape and length of a cable hanging between two level supports, from the span and the sag at the middle. It gives the catenary parameter, the cable length, the extra length over the straight span and the slope at the supports. Use it for washing lines, fence wires, festoon lights, chains and as a calculus example.
How it is calculated
Shape (lowest point at the origin): y = a (cosh(x/a) − 1)
Sag at mid-span: h = a (cosh(L/2a) − 1), solved numerically for a
Cable length: s = 2a sinh(L/2a)
Slope at each support: arctan(sinh(L/2a))
- L is the span between the supports and h the sag; both in metres.
- Sag-to-span ratio = h ÷ L.
The principle
A flexible cable of uniform weight per length can only carry tension along itself. At every point the horizontal part of the tension is the same, while the vertical part grows with the weight of cable hanging below that point. That balance gives a differential equation whose solution is the hyperbolic cosine, the catenary. The parameter a equals the horizontal tension divided by the weight per metre: a large a means a tight, shallow cable; a small a a deep sag. The length follows from the arc-length integral of the curve.
Worked example
Default inputs: span 100 m, sag 10 m.
- Solve 10 = a (cosh(50/a) − 1): a = 126.632 m.
- Cable length = 2 × 126.632 × sinh(50 ÷ 126.632) = 102.619 m.
- Extra length over the span = 102.619 − 100 = 2.619 m.
- Slope at each support = arctan(sinh(0.39484)) = 22.057°.
- Sag-to-span ratio = 10 ÷ 100 = 0.1.
| Span between the two supports (m) | Cable length (m) |
|---|---|
| 25 | 33.6047 |
| 50 | 54.9859 |
| 75 | 78.4449 |
| 100 | 102.619 |
| 125 | 127.108 |
| 150 | 151.763 |
| 200 | 201.327 |
How to read the result
A small amount of extra cable produces a lot of sag: here 2.6 % extra length gives a 10 m dip. The reverse is also true: pulling a cable nearly straight takes very large tension, because a grows and so does the horizontal force. The slope at the supports shows the direction of the pull on each anchor.
Common mistakes
- Using a parabola for large sags; it is a good approximation only for shallow cables.
- Measuring the sag from the wrong reference when supports are at different heights.
- Ignoring cable stretch and temperature changes.
- Confusing the sag with the total drop from support to ground.
Frequently asked questions
What is a catenary?
The curve formed by a uniform flexible chain or cable hanging under its own weight: y = a cosh(x/a).
Is a hanging chain a parabola?
No, but for small sag the two curves are very close.
How long a rope do I need for a given sag?
Enter the span and sag; the cable length line gives it.
Sources: OpenStax, Calculus Volume 1 (2016), §6.9 Calculus of the Hyperbolic Functions