Calculate the height, arc length, slope, curvature, and vertex of a catenary curve from the standard hanging-cable model.
Last updated: June 2026 | By Patchworkr Team
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Parameter a needs a real number.
A catenary is the curve formed by a hanging chain or cable under its own weight. It is not the same as a parabola.
The standard equation is y = a * cosh(x / a), where a controls how tight or flat the curve is.
The lowest point occurs at x = 0, with y = a.
With a = 20 and x = 15, the height is 20 * cosh(15/20) = 22.722... .
The curve rises above its lowest point at x = 0, and the tangent angle matches the slope at that point.
Why must a be positive?
A positive a keeps the model physically meaningful and avoids a flipped or invalid scale factor in this tool.
Can x be negative?
Yes. The catenary is symmetric, so negative x values work just like positive ones.
Is the curvature formula exact?
Yes. For y = a * cosh(x / a), the curvature simplifies to a / y^2.
Does the vertex always stay at x = 0?
In this centered model, yes. The lowest point is at (0, a).
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