Catenary
The hyperbolic-cosine curve y = a·cosh(x/a) of a hanging chain — subtly different from a parabola.
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Formulas and key values
- Cartesian equation
- Arc length · from the lowest point to x
- Lowest point
a is the height of the lowest point, the ratio of horizontal tension to weight per unit length.
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Start the video01 · Definition
What is the Catenary?
Hang a uniform chain from two points and it settles into the catenary y = a·cosh(x/a). Galileo guessed the shape was a parabola; Joachim Jungius disproved that experimentally, and in 1691 Leibniz, Huygens, and Johann Bernoulli each derived the true equation in response to Jakob Bernoulli’s challenge. The name comes from the Latin catena, “chain”.
Flip it upside down and the pure-tension chain becomes a pure-compression arch — Robert Hooke’s insight “as hangs the flexible line, so but inverted will stand the rigid arch”. That is why the Gateway Arch in St. Louis and Gaudí’s hanging-chain models follow (weighted) catenaries rather than parabolas.
02 · Properties
Key properties
- Arc length from the vertex: s = a·sinh(x/a); the sag-to-tension geometry is all in one constant a.
- The horizontal tension component is constant along the chain.
- Near the vertex it is approximated by the parabola y ≈ a + x²/(2a) — the source of Galileo’s confusion.
- Rotating it about the x-axis gives the catenoid, the only non-planar minimal surface of revolution.
- A square wheel rolls smoothly on a road made of inverted catenary humps.
03 · Questions
Catenary: common questions
- What is the equation of the Catenary?
- Cartesian equation: y = a cosh x/a = (a/2)(e^(x/a) + e^(-x/a)). a is the height of the lowest point, the ratio of horizontal tension to weight per unit length.
- What is the arc length of the Catenary?
- Arc length (from the lowest point to x): s = a sinh x/a. a is the height of the lowest point, the ratio of horizontal tension to weight per unit length.
04 · Keep exploring