Catenary Cable Sag Calculator
Enter span and cable weight per metre then solve for sag from tension, or find the tension needed for a given sag. Cable length and SVG catenary curve included. Nothing uploaded.
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Learn more: catenary curves and cable sag
What catenary sag is
A cable or chain hung between two points, carrying nothing but its own weight, settles into a catenary curve. Sag is the vertical drop from the line between the supports down to the lowest point of the cable at midspan.
Sag grows as the cable gets heavier per metre and shrinks as you pull it tighter. That is the relationship behind clearance checks under an overhead wire and behind how hard a zip line pulls on its anchors.
How the calculator gets from tension to sag
Everything follows from one value, the catenary parameter a, which is the horizontal tension divided by the cable's weight per unit length. You enter weight in kilograms per metre and the tool multiplies by 9.81 to convert it to newtons per metre first, so a comes out in metres.
Sag at midspan is then a × (cosh(L/2a) - 1), with L as the span. Cable length is 2a × sinh(L/2a) and tension at the supports is the horizontal tension times cosh(L/2a). The hyperbolic form is used at every sag ratio, with no approximation anywhere in the code.
Working the other way, from a sag you want to a tension you need, the calculator starts from the parabola estimate a = L²/(8f) and refines it with Newton's method until the step falls below a micrometre.
A worked 20 metre span
Take a 20 m span with a cable weighing 0.5 kg/m, held at 200 N of horizontal tension. That weight is 4.905 N/m, so a = 200 / 4.905 = 40.775 m and midspan sag lands at 1.232 m, a sag-to-span ratio of 6.16%.
Crossing that 20 m gap takes 20.201 m of cable, and each support carries 206 N rather than the 200 N acting horizontally. Raise tension to 1000 N and sag drops to 0.245 m with 20.008 m of cable. Drop it to 100 N and sag more than doubles to 2.502 m.
Where the parabola shortcut breaks down
Many quick sag tools use the parabola formula wL²/(8T_h) instead of the hyperbolic one. On the 20 m example above it gives 1.226 m against the catenary's 1.232 m, low by about 0.5%.
That error depends only on the sag-to-span ratio. At a 2% ratio the parabola is off by 0.05%, at 10% it reads 1.3% low, and at 20% it is 4.7% low. FIRGELLI's cable tension page, which runs on the parabolic model, draws its own line at a sag-to-span ratio of 1:8 and puts the error under 3% there, recommending a catenary calculation for anything deeper.
Cable length and support tension
A sagging cable is always longer than the span, and the surplus climbs quickly once the sag ratio grows. The 6.16% case above needs 201 mm of extra cable across 20 m; the 12.51% case at 100 N needs 812 mm.
The horizontal component of tension is the same at every point along the cable. What the supports carry on top of it is the weight of the cable itself, which is why the support figure is always the horizontal tension times cosh(L/2a). The model assumes both supports sit at the same height and that nothing but the cable's own weight is acting on it.
FAQ
Why is tension at the supports higher than horizontal tension?
The horizontal pull is constant along the whole cable, but at the supports the cable also has to hold up its own weight. The total there is the horizontal tension times cosh(L/2a), which turns 200 N into 206 N in the 20 metre example above.
Can I use this for a power line or a zip line?
Yes. Any flexible cable hanging between two fixed points under its own weight follows the same curve. Ice adds weight you can fold into the kilograms-per-metre input, but wind pushes sideways and this calculator does not model it.
What happens if I enter a very small sag?
The solver starts from the parabola estimate, which is already close at small sag, so it converges in a few steps. A small sag also pushes the catenary parameter a to a large value, and the result moves closer to the parabola answer without ever quite matching it.