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A catenary is the exact curve formed by a chain, cable, or rope hanging freely under its own weight between two supports, described by the equation y = a·cosh(x/a), where a is the catenary parameter (controlling how tightly or loosely the curve sags) and the vertex, or lowest point, sits at (0, a) in a coordinate system centered under that lowest point. The shape appears everywhere gravity pulls on a flexible line: sagging power lines and telephone cables, ships' anchor chains, the necklace-like curve of a loosely draped rope, and — famously — the St. Louis Gateway Arch, which was deliberately built as an inverted catenary because that shape distributes structural forces most efficiently along its length.
How it works
Enter the parameter a and two x-values (positions measured horizontally from the curve's lowest point). The calculator finds y at each x using y = a·cosh(x/a), where cosh is the hyperbolic cosine function (cosh(t) = (eᵗ+e⁻ᵗ)/2). It also reports the sag — how much higher the curve is at your chosen x-value than at the vertex, i.e., y(x) − a — and the arc length of the curve between the two x-values, given by the formula a·sinh(x/a) measured from the vertex, where sinh is the hyperbolic sine (sinh(t) = (eᵗ−e⁻ᵗ)/2).
- Enter catenary parameter a.
- Enter x1.
- Enter x2.
- Click Calculate to see your results.
Examples
a = 10, x = 5
y(5) = 10·cosh(0.5) = 10 × 1.1276 ≈ 11.28. Sag = 11.28 − 10 = 1.28. Arc length from the vertex = 10·sinh(0.5) = 10 × 0.5211 ≈ 5.21.
a = 5, x = 3
y(3) = 5·cosh(0.6) = 5 × 1.1855 ≈ 5.93. Sag = 5.93 − 5 = 0.93. Arc length from the vertex = 5·sinh(0.6) = 5 × 0.6366 ≈ 3.18.
a = 20, x = 10
y(10) = 20·cosh(0.5) = 20 × 1.1276 ≈ 22.55. Sag = 22.55 − 20 = 2.55. Arc length from the vertex = 20·sinh(0.5) = 20 × 0.5211 ≈ 10.42.
Who should use it
- Estimating the sag and arc length of a hanging cable, chain, or wire between two supports.
- Physics and engineering coursework on catenary curves.
- Architectural and structural design referencing catenary/arch shapes.
Industry applications
- Civil and structural engineering
- Electrical utility line design
- Physics education
Advantages
- Computes y-value, sag, and arc length together from one input.
- Matches the true physical shape of a freely-hanging cable or chain.
Limitations
- Not the correct model for a cable under significant additional uniform loading (like a suspension bridge deck) — that's closer to a parabola instead.
Common mistakes to avoid
- Confusing a catenary with a parabola — they look nearly identical but follow genuinely different equations and arise from different physical loading conditions.
- Forgetting that x is measured from the vertex (the curve's lowest point), not from one of the cable's two end supports.
- Mixing up sag (height above the vertex) with the y-value itself — sag is y(x) minus a, not y(x) alone.
Best practices
- Double check whether your problem is describing a freely-hanging cable (a true catenary) or a uniformly-loaded cable like a suspension bridge deck (closer to a parabola) before applying this formula.
- Remember that a controls the tightness of the curve — verify your value of a is physically reasonable for the span and sag you expect.
- Keep x measured relative to the vertex, not an arbitrary support point, to avoid sign or offset errors.
Tips
- Working with a cycloid or spiral curve instead? See the Cycloid Calculator or Spiral Length Calculator.