Math
Catenary Curve Calculator
A chain or cable hanging under its own weight between two supports forms a catenary, not a parabola.
Enter the span and the sag (or the parameter a directly) and mathcheck solves for a, writes the equation y = a cosh(x/a), and returns the arc length, the ratio of support tension to horizontal tension, the angle at the supports, and, if you give the weight per unit length, the actual tensions. The graph overlays the parabola with the same span and sag so you can see how close the approximation is.
Supports are assumed to be at the same height.
For example N/m or lb/ft. Leave 0 to skip the force calculations.
Try an example
Result
Catenary parameter a
—
- Equation
- —
- Arc length
- —
- Sag
- —
- Tension ratio T / H
- —
- Angle at the supports
- —
- Forces
- —
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Study path
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Why a hanging cable is a catenary
Balancing the forces on a small piece of a uniform cable shows that its How steep a line is: how much y changes for each step right in x. grows in proportion to the length already hanging, which leads to the hyperbolic cosine. With the lowest point at (0, a), the curve is y = a cosh(x/a), where a = H/w is the horizontal tension divided by the weight per unit length. A parabola arises instead when the load is spread uniformly along the horizontal, as in a suspension bridge deck, so the two curves look alike only for shallow sag.
Span 100, sag 10
Solving 10 = a (cosh(50/a) - 1) numerically gives a ≈ 126.63. The cable length is 2(126.63) sinh(50/126.63) ≈ 102.62, the support tension is 1.079 times the horizontal tension, and the cable meets each support at about 22.1° from horizontal.
| Sag / span | Catenary vs. parabola | Support angle |
|---|---|---|
| under 5% | nearly identical | shallow, under 12° |
| 10% | differ by about 1% in sag | about 22° |
| 50% | clearly different shapes | about 68° |
| over 100% | deep loop; parabola useless | nearly vertical |
How to use this calculator
- Choose whether you know the span and sag or the parameter a.
- Enter the span (and sag or a). Add the cable's weight per unit length if you want actual tensions.
- Read a, the equation, the cable length, the tension ratio, and the support angle.
- Compare the parabola approximation line with the exact values to judge whether the simpler curve would do.
- Open Show the work to see the numerical solve, the check, and each formula evaluated.
How to read the answer
A larger a means a tighter, flatter cable with higher horizontal tension. The arc length is the amount of cable needed. The tension ratio shows how much more the supports carry than the The point exactly halfway between two other points.; it grows quickly as the sag shrinks, which is why a nearly straight cable needs enormous tension. The support angle is the direction the cable pulls on its anchor. Forces appear only when a weight per unit length is provided.
Common mistakes and edge cases
- Using a parabola for a deep-sag cable. The shapes diverge once the sag is more than about a tenth of the span.
- Confusing a with the sag or the height of the supports; a is the height of the lowest point above the x-axis in the standard placement.
- Assuming the horizontal tension changes along the cable. H is constant; only the vertical component and total tension grow toward the supports.
- Entering supports at different heights. This page assumes level supports; an unequal case needs a shifted The turning point of a parabola, or a corner point of a shape..
- Reporting the span as the cable length. The cable is always longer than the span.
Worked examples
Power line
Span 100, sag 10
Catenary parameter a
126.632436
With cable weight
Span 100, sag 10, w = 2 per unit length
Catenary parameter a
126.632436
Shallow sag (parabola-like)
Span 200, sag 2
Catenary parameter a
2,500.333262
Deep loop
Span 10, sag 20
Catenary parameter a
1.486902
From the parameter
a = 50, span 100
Catenary parameter a
50
Zero sag
Catenary parameter a
—
Zero span
Catenary parameter a
—
Frequently asked questions
What is the equation of a catenary?+
y = a cosh(x/a), where cosh is the hyperbolic cosine and a is the horizontal tension divided by the weight per unit length. The lowest point is at (0, a); subtracting a puts it at the origin.
Is a hanging chain a parabola?+
No. A chain under its own weight is a catenary. A parabola describes a cable whose load is uniform per horizontal distance, like a suspension bridge with a heavy deck. For shallow sag the two are almost indistinguishable.
How do I find a from the span and sag?+
Solve s = a(cosh(L/2a) - 1). There is no algebraic solution, so it is done numerically; the calculator bisects until the equation balances.
How long is the cable?+
The arc length is 2a sinh(L/2a). For shallow sag this is close to L + 8s^2/(3L).
Where is the tension greatest?+
At the supports, where T = w(a + s). At the lowest point the tension is purely horizontal and equals H = w a. The ratio is cosh(L/2a).
What if the supports are at different heights?+
The curve is still y = a cosh(x/a), but the lowest point shifts sideways and the two halves have different lengths. This page assumes level supports; for unequal heights, treat each side separately with its own half-span.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026