Catenary
Work out a catenary, the curve of a chain hanging from two points, from its span and sag, or turn it upside down for a catenary arch from its span and rise. You can also give the span and the length of the chain. The tool solves y = a cosh(x/a), shows the length of the curve, the angle at the ends and how far a parabola of the same size would be off, and prints the curve at real size across as many sheets as it needs, with an offset table (height at every step along the base) so you can mark it out on a board, wall or floor. Download a PDF, PNG, SVG or CSV. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints across as many sheets as it needs: cut each sheet on its dashed line and butt the edges together, matching the corner marks. The SVG holds the whole drawing at real size.
Catenary
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Runs in your browser with our own code: nothing you type is uploaded or saved. Every line and number is worked out from y = a cosh(x/a) by our own code. This is geometry only, not structural advice. You may use what you make here, including commercially, with no credit needed.
How to draw a catenary and print it at real size
- Pick a preset, or choose Hanging or Arch and type the span and the sag (or rise), or the span and the length of the curve.
- Tick the parabola to compare the two curves, and the offset stations to see where the table measures.
- Set the offset step (for example every 50 mm or every 2 in).
- Choose Fitted to one page for a diagram, or Real size for a full-size template on tiled sheets.
- Download the PDF (the offset table is on its own page) and print at 100% (Actual size), or the SVG or the CSV of offsets.
What you can use it for
- Marking out an arch shape on plywood, card or a wall
- Garden arches, doorway and niche shapes for models and crafts
- Drapes, swags and chain or bead loops for jewellery and decor
- Maths and physics lessons: the hanging chain and the hyperbolic cosine
- Comparing a catenary with a parabola of the same size
- Pen plotter and laser cutting of exact curves
The 14 presets
| Preset | Shape | Span | Sag or rise | a | Curve length | Gap to a parabola |
|---|---|---|---|---|---|---|
| Hanging chain: 1 m span, 250 mm sag | Hanging | 1000 mm | 250 mm | 537.2 mm | 1150.8 mm | 4.4 mm |
| Necklace drape: 200 mm span, 80 mm sag | Hanging | 200 mm | 80 mm | 72.9 mm | 268.8 mm | 2.9 mm |
| Shallow sag: 2 m span, 100 mm sag | Hanging | 2000 mm | 100 mm | 5016.6 mm | 2013.3 mm | 0.1 mm |
| Deep loop: 300 mm span, 450 mm sag | Hanging | 300 mm | 450 mm | 50.2 mm | 995.3 mm | 64.6 mm |
| Rope 1.5 m long between posts 1.2 m apart | Hanging | 1200 mm | 398.2 mm | 507.3 mm | 1500 mm | 11.1 mm |
| Garden arch: 1200 mm span, 2100 mm rise | Arch | 1200 mm | 2100 mm | 188.1 mm | 4560.7 mm | 332.7 mm |
| Doorway arch: 36 in span, 80 in rise | Arch | 36 in | 80 in | 5.14 in | 169.98 in | 14.51 in |
| Window head: 24 in span, 12 in rise | Arch | 24 in | 12 in | 7.43 in | 35.9 in | 0.6 in |
| Shelf niche: 400 mm span, 600 mm rise | Arch | 400 mm | 600 mm | 66.9 mm | 1327 mm | 86.2 mm |
| Low arch: 1 m span, 200 mm rise, with the parabola | Arch | 1000 mm | 200 mm | 655.9 mm | 1099.7 mm | 2.4 mm |
| Tall arch: 300 mm span, 900 mm rise | Arch | 300 mm | 900 mm | 38.6 mm | 1875.7 mm | 188.5 mm |
| Rise equal to span: 50 cm × 50 cm | Arch | 50 cm | 50 cm | 10.14 cm | 118.55 cm | 5.27 cm |
| Table-top model arch: 150 mm span, 150 mm rise (real size) | Arch | 150 mm | 150 mm | 30.4 mm | 355.7 mm | 15.8 mm |
| Arch of a given length: 600 mm curve over a 300 mm span | Arch | 300 mm | 238.9 mm | 68.9 mm | 600 mm | 20.4 mm |
Every value is worked out by the tool from the span and the sag, rise or length. The gap is the largest vertical distance between the catenary and a parabola with the same span and depth.
Depth, length and the parabola gap
| Depth / span | a | Curve length | End angle | Parabola gap | Gap as % of depth |
|---|---|---|---|---|---|
| 0.05 | 2.5083 L | 1.0066 L | 11.3° | 0.00004 L | 0.1% |
| 0.1 | 1.2663 L | 1.0262 L | 22.1° | 0.00032 L | 0.3% |
| 0.25 | 0.5372 L | 1.1508 L | 47.0° | 0.00439 L | 1.8% |
| 0.5 | 0.3094 L | 1.4958 L | 67.5° | 0.02503 L | 5.0% |
| 1 | 0.2027 L | 2.3710 L | 80.3° | 0.10540 L | 10.5% |
| 2 | 0.1487 L | 4.2871 L | 86.0° | 0.34253 L | 17.1% |
L is the span. Shallow curves are close to a parabola; deep ones are not. The end angle is measured from level at each support.
An offset table: 1 m span, 300 mm sag or rise, every 100 mm
| Distance from the left end | Height (arch) or drop (chain) |
|---|---|
| 0 mm | 0.0 mm |
| 100 mm | 114.6 mm |
| 200 mm | 198.5 mm |
| 300 mm | 255.8 mm |
| 400 mm | 289.1 mm |
| 500 mm | 300.0 mm |
| 600 mm | 289.1 mm |
| 700 mm | 255.8 mm |
| 800 mm | 198.5 mm |
| 900 mm | 114.6 mm |
| 1000 mm | 0.0 mm |
The tool makes this table for any span, depth and step, prints it on its own PDF page and saves it as a CSV.
What prints
| Download | What is in it |
|---|---|
| The drawing fitted to one page or at real size on tiled sheets, then the offset table page(s); vector, with a check bar on every sheet | |
| PNG 300 DPI | The same sheets as pictures (a ZIP when there is more than one) |
| SVG (real size) | Only the drawing at real size: the curve as one path, guides as separate paths |
| Offsets CSV | The offset table for a spreadsheet |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| A hanging chain or catenary arch, its length and a real-size template | Catenary (this page) |
| An evenly spaced spiral | Archimedean Spiral |
| A flat pattern for a cone or a lampshade | Cone Template |
| Gears and gear curves | Gear Generator |
| Squared paper for drawing your own curves | Graph Paper |
Other drawing tools: the Cone Template makes flat patterns, the Gear Generator draws spur gears and the Graph Paper maker prints grids at real size.
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Frequently asked questions
What is a catenary?
The curve that a chain or rope of even weight takes when it hangs freely from two points. Its equation is y = a cosh(x/a), where a sets how flat it is. The word comes from the Latin catena, chain.
What is a catenary arch?
The same curve turned upside down. On this page the arch shape is exactly the hanging curve flipped, with the same span and the rise in place of the sag. This page gives the geometry only; it does not size or check any structure.
How does the tool find a from the span and the sag?
With span L and sag s it solves (cosh u − 1)/(2u) = s/L for u by repeated halving, then a = L/(2u). Our tests check that the solved curve gives back the typed sag (or rise) and span to within one billionth.
How long is the curve?
The length is 2a sinh(L/2a). For a span of 1 m and a sag of 300 mm, a = 459.4 mm and the curve is 1209.4 mm long. Our tests check the formula against a numerical integration to one millionth.
Is a catenary the same as a parabola?
No. With the same span and depth the two curves differ, and the gap grows as the curve gets deeper: the table below gives it for several shapes, and the tool shows it for yours.
What is the offset table?
A list of heights (for an arch, above the springing line; for a chain, the drop below the line between the supports) at every step along the base, measured from the left end. Mark each one and join the marks with a flexible batten. The values come from the same formula as the drawing and are rounded to 0.1 mm or 0.01 in.
Can I set the length of the chain instead of the sag?
Yes. Choose Span and the length of the curve. The tool solves sinh(u)/u = length / span for u, then the sag follows. The length must be longer than the span.
How does the real-size print work when the drawing is bigger than the paper?
Choose Real size. The tool splits the drawing over as many sheets as it needs, each with a dashed edge, corner marks and a label such as A1 (row A, column 1). Cut each sheet on its dashed line, butt the edges together in order and tape them. Every printed point is at its true position to within 0.2 mm in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar on each sheet lets you confirm the scale.
Can I use it on a plotter or cutter?
Yes. The SVG holds the curve at real size as one path, with the base line, the centre line and any parabola or stations as separate paths. For a program that wants DXF, convert the SVG with our SVG to DXF tool.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every line and number is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.