Catenary Curve Calculator
Solve catenary curve problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Catenary Curve Calculator
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Formula: y = a * cosh(x / a)
Worked example — Sag: 5.4308 m | Arc Length: 23.5040 m | Support Tension: 15.4308 (normalized)
Formula
y = a * cosh(x / a)
The catenary equation describes the curve of a hanging chain or cable under uniform gravity, where a is the catenary parameter (horizontal tension divided by weight per unit length), x is the horizontal distance from the lowest point, and cosh is the hyperbolic cosine function. Arc length is s = a * sinh(x/a).
Worked Examples
Example 1: Power Line Catenary Analysis
Problem:A power line spans 20 meters with a catenary parameter a = 10. Find the sag, total cable length, and tension at the support.
Solution:Sag = a * (cosh(L/(2a)) - 1) = 10 * (cosh(10/10) - 1) = 10 * (cosh(1) - 1) = 10 * (1.5431 - 1) = 5.4308 m Total arc length = 2a * sinh(L/(2a)) = 20 * sinh(1) = 20 * 1.1752 = 23.5040 m Tension at support (normalized) = a * cosh(L/(2a)) = 10 * cosh(1) = 10 * 1.5431 = 15.4308
Result:Sag: 5.4308 m | Arc Length: 23.5040 m | Support Tension: 15.4308 (normalized)
Example 2: Slope and Curvature at a Point
Problem:For a catenary with a = 10, find the slope, angle, and radius of curvature at x = 5.
Solution:Slope = sinh(x/a) = sinh(5/10) = sinh(0.5) = 0.5211 Angle = arctan(0.5211) = 27.5478 degrees Radius of curvature = a * cosh^2(x/a) = 10 * cosh^2(0.5) = 10 * (1.1276)^2 = 10 * 1.2715 = 12.7153
Result:Slope: 0.5211 | Angle: 27.5478 deg | Radius: 12.7153
Frequently Asked Questions
What is a catenary curve and how is it formed?
A catenary curve is the shape that a flexible, inextensible chain or cable assumes when supported at its endpoints and acted upon solely by gravity. The word catenary comes from the Latin word catena meaning chain. The mathematical equation of a catenary is y = a * cosh(x/a), where a is the catenary parameter equal to the horizontal tension divided by the weight per unit length, and cosh is the hyperbolic cosine function. Despite looking similar to a parabola, a catenary is fundamentally different. Galileo initially mistook it for a parabola, but Leibniz, Huygens, and Johann Bernoulli independently proved its true form in 1691 using the newly developed calculus.
What is the catenary parameter and what does it represent?
The catenary parameter, typically denoted as a, is the ratio of horizontal tension (T_h) to the weight per unit length (w) of the cable: a = T_h / w. It has units of length and determines the shape and tightness of the catenary. A large value of a produces a flatter, more taut curve with higher tension, while a small value of a produces a more deeply sagging curve with lower tension. At the lowest point of the catenary (x = 0), the parameter a equals both the y-coordinate (height above the directrix) and the radius of curvature. The catenary parameter is the single most important value characterizing a catenary because the entire shape is determined once a and the support positions are known.
How does a catenary differ from a parabola?
Although a catenary and a parabola look similar for shallow curves, they are mathematically distinct. A parabola is described by y = x^2/(2a) while a catenary is described by y = a * cosh(x/a). The key physical difference is that a catenary describes a hanging chain with uniform weight per unit arc length (uniform density cable), while a parabola describes a cable with uniform weight per unit horizontal length (like a suspension bridge deck distributing load evenly along the horizontal span). For small sag-to-span ratios (less than about 1:8), a catenary closely approximates a parabola. As the sag increases, the differences become more pronounced, with the catenary sagging less in the middle but having steeper sides than the equivalent parabola.
How do you calculate the sag of a catenary cable?
The sag of a catenary cable is the vertical distance between the lowest point and the support points. For a catenary y = a * cosh(x/a) with supports at x = plus or minus L/2 (span = L), the sag equals a * cosh(L/(2a)) - a = a * (cosh(L/(2a)) - 1). For a given span, the sag depends entirely on the catenary parameter a. Increasing tension (larger a) reduces sag, while decreasing tension (smaller a) increases sag. A useful approximation for small sag is: sag is approximately L^2 / (8a), which becomes exact for a parabola. In power line design, typical sag-to-span ratios range from 2 to 5 percent, and precise sag calculation is critical for maintaining safe clearances.
What is the arc length formula for a catenary?
The arc length of a catenary from the lowest point to a horizontal distance x is given by s = a * sinh(x/a), where sinh is the hyperbolic sine function. The total arc length between supports at x = plus or minus L/2 is 2a * sinh(L/(2a)). This elegant formula reflects a beautiful property of the catenary: the arc length has a simple closed-form expression involving only the catenary parameter and the horizontal distance. The total cable length is always greater than the span, and the excess length beyond the span directly relates to the sag. For power line engineers, calculating the exact cable length is essential for material procurement and installation planning, as too much or too little cable causes either excessive sag or excessive tension.
How is the catenary used in architecture and engineering?
The catenary shape is fundamental in architecture and engineering because of its unique structural properties. An inverted catenary forms the ideal arch shape where all forces are compressive with no bending moments, making it the most efficient form for masonry arches. Antonio Gaudi famously used hanging chain models inverted to design the arches and vaults of the Sagrada Familia and Casa Mila. The Gateway Arch in St. Louis is a weighted catenary (modified to account for varying cross-section). In cable engineering, catenary calculations determine the sag, tension, and clearance of power transmission lines, suspension bridges, and cable-stayed structures. The catenary also appears in the design of cooling tower shells and tent structures.
How do temperature and wind affect catenary cable shapes?
Temperature and wind significantly alter catenary cable behavior in real-world applications. Temperature changes cause thermal expansion or contraction of the cable, changing its effective length: as temperature rises, the cable elongates, increasing sag, and as temperature drops, the cable shortens, increasing tension and reducing sag. Power line engineers must design for extreme temperature ranges, as summer heat can increase sag enough to violate minimum ground clearance requirements. Wind loads add a horizontal force component that changes the curve from a catenary in a vertical plane to a three-dimensional curve. The effective weight per unit length becomes the vector sum of gravitational and wind forces, typically modeled as an increased effective weight.
What is the tension distribution along a catenary cable?
The tension in a catenary cable varies along its length. At the lowest point, the tension is purely horizontal and equals T_h = w * a (weight per unit length times the catenary parameter). As you move toward the supports, the tension increases because it must support the weight of cable below plus maintain the horizontal component. At any point x, the tension is T(x) = w * a * cosh(x/a) = w * y(x), meaning the tension at any point equals the weight per unit length times the height of that point above the directrix. The maximum tension occurs at the support points where the cable angle is steepest. The minimum tension at the lowest point equals the horizontal component T_h, and the vertical component varies from zero at the bottom to maximum at the supports.
What is the radius of curvature of a catenary?
The radius of curvature of a catenary at any point x is R = a * cosh^2(x/a). At the lowest point (x = 0), the radius of curvature equals a, the catenary parameter itself. This means the catenary is most sharply curved at its lowest point and becomes flatter (larger radius of curvature) as you move toward the supports. The radius of curvature is important in cable engineering because it affects bending stresses in the cable. For cables with significant bending stiffness (such as thick wire ropes), the actual shape deviates from a pure catenary near the support points where the radius of curvature is large and bending effects become negligible, but differs near attachment points where the cable must conform to the support geometry.
How do you solve for the catenary parameter when given span and cable length?
When the span (L) and total cable length (S) are known, finding the catenary parameter a requires solving the transcendental equation S = 2a * sinh(L/(2a)). This equation has no closed-form solution and must be solved numerically using methods such as Newton-Raphson iteration or bisection search. Starting with an initial estimate a0 = L^2 / (8 * sag) where sag is approximated from the excess cable length, Newton method converges rapidly. The equation can also be rearranged as sinh(L/(2a)) = S/(2a) and solved iteratively. Commercial engineering software handles this automatically, but understanding the underlying numerical process is important for verifying results and handling special cases where convergence may be slow.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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