Calculate properties of a catenary, the curve formed by a hanging chain or cable. Essential for architecture, bridge design, and engineering.
Last updated: April 2026 | By Patchworkr Team
Shape parameter (vertical scale)
Distance from center
A catenary is the curve that a flexible chain or cable assumes when supported at its ends and acted upon by gravity. The name comes from the Latin word *catena*, meaning "chain."
Key characteristics:
Catenaries appear in suspension bridges, power lines, arches, and anywhere a flexible cable hangs under its own weight.
Identify or calculate the parameter 'a', which physically equals H/w (horizontal tension divided by weight per unit length). Why: The parameter 'a' completely defines the catenary's shape. Larger 'a' means a flatter, less sagging cable; smaller 'a' means more pronounced sag. You must know this to proceed.
Place x=0 at the center (lowest point of the curve). The vertex (minimum) always occurs at coordinates (0, a). Positive x represents one side; negative x the other (the curve is symmetric). Why: Centering at x=0 simplifies calculations and aligns with standard catenary notation. The axis of symmetry ensures the mathematics works consistently.
Select the horizontal distance from the center where you want to calculate the curve values. This x-value can be positive or negative. Why: Different x positions give different heights and slopes. Measuring at various distances lets you map out the full catenary shape (e.g., for bridge cable placement).
Plug in your 'a' and 'x' values to find the height y. The cosh (hyperbolic cosine) function naturally models how gravity and tension shape the curve. Why: This equation comes directly from the physics: the curve minimizes potential energy under the tension-weight balance.
From y, derive the arc length s = a·sinh(x/a), slope dy/dx = sinh(x/a), and curvature κ = 1/y to understand the cable's local behavior. Why: Engineers need these properties for stress analysis, clearance planning, and structural design. The slope tells you angle; arc length tells you cable needed; curvature tells you how tightly the curve bends.
Power Line Design
y = a · cosh(x/a), where 'a' is a parameter that controls the curve's shape, and cosh is the hyperbolic cosine function.
No. They look similar, but a catenary is flatter at the bottom and steeper at the sides. A parabola has constant curvature change, while a catenary's curvature varies.
Hanging chains, power lines, suspension bridge cables, decorative arches, and anywhere a flexible cable hangs under its own weight.
The Gateway Arch in St. Louis is an inverted catenary—the optimal shape for a freestanding arch to support its own weight.
The physics of hanging cables naturally leads to hyperbolic functions. They describe the balance between tension and weight.
Physically, a = H/w, where H is horizontal tension in the cable and w is weight per unit length. Larger 'a' means less sag.
Yes! Suspension bridge cables follow catenaries when supporting only their own weight. With added load (roadway), they approach parabolas.
The minimum height occurs at x = 0 and equals the parameter 'a'. This is the vertex of the curve.
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