Catenary Curve Calculator - Plot, Tabulate, and Evaluate the Hanging-Chain Curve
Use our free catenary curve calculator to plot y = a cosh(x/a) for a hanging chain or cable. Pick the standard or weighted form, set the sag parameter a, and read off y(x), the vertex, and the arc length.
Catenary Curve Calculator
Results
What Is a Catenary Curve Calculator?
A catenary curve calculator is a focused math tool that plots and evaluates the curve a perfectly flexible chain or cable forms when it hangs under its own weight. The curve is described by y = a · cosh(x / a), where a is the sag parameter that controls how tightly the chain opens, and cosh is the hyperbolic cosine. This catenary curve calculator updates the (x, y) table, the SVG graph, and the result panel together as you change any input.
- • Physics and statics problems: Work out the exact shape and length of a hanging cable before you buy material or design supports.
- • Architecture and arch design: Sanity-check that a proposed arch, dome, or gateway matches a true catenary rather than a near-identical parabola.
- • Suspension bridge estimates: Compare a true catenary with the parabolic shape engineers use for road cables, and quantify how close the two curves really are.
- • Math homework and teaching: Generate tabulated (x, y) pairs, an SVG graph, and the arc length for a clear classroom demonstration of hyperbolic functions.
If you want to compare the catenary with a closely related curve, our parabola calculator covers the quadratic form y = a (x - h)^2 + k with vertex and focus outputs.
Because the formula uses hyperbolic cosine, the catenary is also a good companion to exponential work; our exponential notation calculator handles the e^x and e^-x pieces inside cosh(x / a).
How the Catenary Curve Calculator Works
The catenary curve calculator evaluates the catenary equation at a chosen x, then re-evaluates it across your domain at the step you specify to build the table and the SVG graph. The vertex is at x = 0, and the arc length is the integral of sqrt(1 + (dy/dx)^2) over [xMin, xMax], which has the closed form a · (sinh(xMax / a) - sinh(xMin / a)). Every keystroke reruns the catenary curve calculator so the table, graph, and result panel stay in sync.
- a (sag parameter): Sets the horizontal scale. Larger |a| gives a flatter, broader curve; smaller |a| makes the curve narrower. The unit matches the unit of x.
- b (weight parameter): Scales the curve vertically. Only active in weighted mode; ignored when the standard form is selected.
- x: Horizontal position measured from the vertex. The catenary is symmetric about x = 0, so positive and negative x give mirror-image arms.
- y: Vertical position above the vertex. The vertex minimum is a (standard) or b (weighted).
Pair the closed-form arc length above with our arc length calculator for the analogous s = rθ on a circular arc.
According to Wolfram MathWorld, the catenary is the curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force, and its equation y = a cosh(x / a) was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691.
Standard catenary, a = 1, on x in [-2, 2]
Inputs: mode = standard, a = 1, b = 1, xMin = -2, xMax = 2, step = 0.5, evalX = 0.5.
1. y at x = 0.5: cosh(0.5) = (e^0.5 + e^-0.5) / 2 ≈ 1.1276, so y(0.5) = 1 · 1.1276 = 1.1276.
2. Vertex at x = 0: y(0) = 1 · cosh(0) = 1.0.
Result: y(0.5) ≈ 1.1276, vertex y = 1.0, arc length ≈ 7.2537 over [-2, 2].
The vertex sits at (0, 1) and the chain rises smoothly to about y(2) ≈ 3.7622. The arc length is roughly 1.81 times the domain width because the curve is longer than its horizontal projection.
Weighted catenary, a = 1.5, b = 0.8, on x in [-3, 3]
Inputs: mode = weighted, a = 1.5, b = 0.8, xMin = -3, xMax = 3, step = 0.5, evalX = 0.
1. y at x = 0: 0.8 · cosh(0) = 0.8 · 1 = 0.8.
2. Vertex at x = 0: y(0) = 0.8.
Result: y(0) = 0.8, vertex y = 0.8, arc length ≈ 10.8806 over [-3, 3].
The b = 0.8 weight pulls the whole curve down by 20% relative to a = 1, while a = 1.5 widens it. The arc length grows faster than the domain width because cosh flares up at the edges.
Key Concepts Explained
These four ideas are the foundation of every catenary problem. Knowing them lets you read the calculator's output as a real physical or geometric quantity, not just a formula.
Sag parameter a
The single most important number in the catenary. It sets the horizontal scale of the curve and is sometimes called the catenary constant. For a physical chain, a equals the horizontal tension divided by the chain's weight per unit length.
Hyperbolic cosine cosh
An even function defined as (e^u + e^-u) / 2. It is the hyperbolic cousin of the regular cosine and is the only elementary function whose graph describes a hanging chain.
Vertex and axis of symmetry
The vertex is the lowest point of the curve at x = 0. The catenary is symmetric about the vertical line x = 0, so a positive arm mirrors the negative arm exactly.
Arc length formula
The length of the curve between x = xMin and x = xMax is a · (sinh(xMax / a) - sinh(xMin / a)). This is one of the few plane curves whose arc length has a clean closed form.
The catenary's defining property is that the slope at any point is proportional to the arc length from the vertex. This is why, for any reasonable chain or cable, the curve looks the same on both sides of the lowest point.
How to Use This Calculator
Follow these five steps to plot the catenary, fill out a table, and get the arc length for a domain of your choice.
Choose a catenary type
Pick Standard for y = a cosh(x / a), or Weighted for y = b cosh(x / a).
Set the sag parameter a
Enter a value in the same unit as x. Try a = 1 to start; larger values flatten the curve.
Set the domain and step
Fill in xMin, xMax, and step h. The table will include both endpoints and every step in between.
Pick the evaluation point
Type the x value at which you want y(x) returned in the primary result box.
Read the result and graph
The result panel reports y(x), the vertex minimum, and the arc length. The SVG below the form shows the curve and the (x, y) sample points.
If you enter a = 1, xMin = -2, xMax = 2, and step = 0.5, the table includes x = -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2 with y = a cosh(x / a) at each point. The result panel reports y(0.5) ≈ 1.1276, vertex y = 1.0, and arc length ≈ 7.2537.
For the inverse perspective, our anti-logarithm calculator evaluates log base 10 and natural log so you can recover a from a measured arc length.
Benefits of Using This Calculator
- • Direct (x, y) table: You get a clean, rounded table for the whole domain, not just a single point, so you can copy values into a spreadsheet or a homework answer.
- • Live SVG graph: The plot updates as you type, so changing a, b, or the domain immediately shows you how the curve responds. No reload, no separate graphing step.
- • Arc length without integration: The arc length over your domain is computed from the closed-form formula a · (sinh(xMax / a) - sinh(xMin / a)). You avoid a manual hyperbolic trig integral.
- • Standard and weighted modes: Switch between the textbook catenary y = a cosh(x / a) and the vertical-stretch variant y = b cosh(x / a) without leaving the page.
- • Validation against Wolfram MathWorld: The formula and the worked example match the MathWorld reference, so the values you read off are consistent with a published source.
Factors That Affect Your Results
Three factors dominate how the catenary behaves. Adjusting them changes the shape, the vertex height, and the arc length in predictable ways. The catenary curve calculator recomputes y(x), the vertex, and the arc length together so you can compare curves side by side.
Sag parameter a
Larger |a| flattens and widens the curve; smaller |a| sharpens it. Doubling a does not double the y values because cosh(x / a) is non-linear in a.
Weighted parameter b
Multiplies the whole curve. b = 2 doubles every y and the arc length stays the same; b = 0.5 halves them. Useful for vertically stretching a known catenary.
Domain width and step
Wider domains increase the arc length and let the curve rise to higher y at the edges. Smaller steps produce smoother plots and longer tables.
- • The model assumes a uniform, perfectly flexible chain under gravity. Real cables with stiffness or point loads deviate from a true catenary.
- • The arc length is computed with the analytic formula for a continuous curve, not the sum of straight-line segments between sample points.
- • a = 0 is mathematically undefined. The calculator returns zeros and shows a guidance message rather than attempting a division by zero.
According to the MacTutor History of Mathematics Archive, the catenary is the focus locus of a parabola rolling along a straight line, and Jungius in 1669 disproved Galileo's earlier claim that a hanging chain would form a parabola.
Compare against our circle length calculator, which gives s = 2πr and s = rθ for a closed circular loop of the same span.
Frequently Asked Questions
Q: What is a catenary curve?
A: A catenary is the curve a perfectly flexible, uniform chain or cable forms when it hangs freely from two supports under gravity. Mathematically, it is the graph of y = a cosh(x / a), where a is the sag parameter and cosh is the hyperbolic cosine.
Q: How do I calculate the catenary curve?
A: Pick a value for the sag parameter a and substitute it into y = a cosh(x / a). The vertex is always at (0, a). To get the arc length across a domain, evaluate a · (sinh(xMax / a) - sinh(xMin / a)).
Q: What is the difference between a catenary and a parabola?
A: In a catenary, the chain's weight is uniform along its length, so the curve is described by a hyperbolic cosine. In a parabola, the load is uniform along the horizontal axis, giving the quadratic y = a x^2. The two curves look similar for small x but diverge for x much larger than a.
Q: What is the catenary parameter a?
A: The catenary parameter a sets the horizontal scale of the curve. For a real hanging chain, a equals the horizontal component of the tension divided by the chain's weight per unit length. Larger |a| gives a flatter curve; smaller |a| gives a sharper curve.
Q: Where is the catenary curve used?
A: Catenary shapes show up in suspension bridge cables, overhead power lines, the Gateway Arch in St. Louis, the dome of Brunelleschi's Florence Cathedral, and even the curve an egg makes on a flat surface. The shape is favored wherever a structure needs to carry weight evenly to two supports.
Q: What is the catenary arc length formula?
A: The arc length of y = a cosh(x / a) from x = xMin to x = xMax is a · (sinh(xMax / a) - sinh(xMin / a)). For a symmetric domain [-L, L] this simplifies to 2 a sinh(L / a).