Catenary Length Calculator
Catenary chain length and sag from curve parameter and support span.
What it does
Chains, cables, and spider silk all sag into the same curve when their own weight shapes them, the catenary y = a x cosh(x / a). Its length and dip follow one constant a plus the span, with no other inputs.
The tool above takes the curve constant with the support span. It returns total chain length, sag below the supports, and half length for one side.
The default a = 10 m over a 20 m span hangs 23.5 m of chain with 5.43 m of sag. The imperial mirror of 32.81 ft over 65.62 ft reads 77.12 ft with 17.82 ft of sag.
Curves follow University of Virginia statics notes with Stony Brook catenary pages behind them. Both derive arc length a x sinh(x / a) from the vertex used here.
How to use
- Pick meters or feet.
- Enter the curve constant a.
- Enter the support span.
- Read chain length first.
- Read sag below supports beside it.
- Use half length for one side ordering.
Formula
- a
- Curve constant: high a means shallow curve.
- span
- Distance between the two supports.
- sinh
- Hyperbolic sine of half span over a.
- cosh
- Hyperbolic cosine for the support height.
The vertex sits at midspan with y measured up from it, so half length is a x sinh(span / 2a). Arguments past 50 reject as extreme. Imperial feet convert to meters first (x 0.3048), then display converts back.
Worked example
Example: a = 10, span 20
Metric defaults: curve constant 10 m, support span 20 m.
- Parameter a = 10 m, span 20 m, half span 10 m.
- Ratio: 10 / 10 = 1.
- sinh(1) = 1.17520119, cosh(1) = 1.54308063.
- Half length: 10 x 1.17520119 = 11.752 m.
- Total: 2 x 11.752 = 23.5 m.
- Sag: 10 x 1.54308063 - 10 = 5.43 m.
- Tool shows 23.5 m, 5.43 m, 11.75 m half.
Reference table
| Span | Length | Sag |
|---|---|---|
| 10 m | 10.42 m | 1.28 m |
| 20 m | 23.5 m | 5.43 m |
| 30 m | 42.59 m | 13.52 m |
Limits
- Free hanging uniform chains only.
- Elastic stretch sits outside the math.
- Point loads break the pure curve.
- Inverse solves need numeric root finding.
Common mistakes
- Typing span as half span.
- Using parabolas for chains.
- Setting a to zero.
- Mixing span and length as inputs.
- Ignoring support height in installs.
Try it yourself
Quick checks. Answers sit tucked under each question.
a = 5, span 10?
Half ratio 1 again: length 11.75 m, sag 2.72 m, half 5.88 m.
a = 10, span 10?
Ratio 0.5: length 10.42 m, sag 1.28 m, half 5.21 m.
a = 10, span 30?
Ratio 1.5: length 42.59 m, sag 13.52 m, half 21.29 m.
Questions
What is a catenary?
A freely hanging chain under its own weight. Bridges and power lines hang near it when load spreads evenly.
Is a catenary a parabola?
No. Parabolas suit uniform horizontal loads; chains follow cosh. Swapping them misstates length and sag.
What are the working formulas?
Length = 2 x a x sinh(span / 2a), sag = a x cosh(span / 2a) - a.
Why does the tool reject my inputs?
Raise a or shorten the span. Tiny a against a wide span sags extremely and overflows the math.
Can I fix length instead of a?
Solve length = 2 x a x sinh(span / 2a) for a numerically, then sag follows. That inverse sits outside this page.
What happens with a huge a?
Yes. As a grows, the curve nears a straight line and length nears the span.