Catenary Calculator
Calculate the catenary curve of a chain or cable hanging between two supports. Find the sag, arc length, and tension ratio from the parameter or arc length, with a drawn curve.
Input
Enter the parameter a and span, or the arc length and span, to compute the sag, total arc length, and tension ratio, and draw the catenary.
a = H / w (horizontal tension over weight per unit length)
Horizontal distance 2L between the two supports.
Result
Sag (vertical droop)
5.43080635
length unit
Arc length
23.50402387
Tension ratio (support / lowest point)
1.54308063
Arc length / span
1.17520119
Parameter a
10
Catenary shape
Catenary y=a cosh(x/a). Sag d=a(cosh(L/a)−1), arc length s=2a sinh(L/a), tension ratio cosh(L/a), where L is the half span.
How it works
- A catenary is the curve formed by a uniform chain or cable hanging under its own weight. With the lowest point as the baseline it is y=a cosh(x/a), where a is the parameter equal to H over w, the horizontal tension divided by the weight per unit length.
- With the horizontal distance between supports (the span) equal to 2L and a half span L, the sag is d=a(cosh(L/a)−1) and the total arc length is s=2a sinh(L/a).
- The tension ratio T_end/H=cosh(L/a) divides the maximum tension at a support by the horizontal tension at the lowest point. Tension is smallest at the lowest point and equals H there.
- In arc length mode, sinh(u)/u=s/(2L) with u=L/a is solved by bisection to recover a. The arc length must be longer than the span.
- A parabola is sometimes used as an approximation, but the exact shape of a chain hanging under its own weight is a catenary. All lengths use the same unit you enter.
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Catenary Calculator