Archimedean Spiral
Draw an Archimedean spiral, the spiral whose turns are all the same distance apart: r = a + bθ. Type any three of the inner radius, the outer radius, the number of turns and the gap between turns, and the tool works out the fourth, the length of the line, and draws it with one or two arms, turning either way. Print it at real size as a template (tiled over sheets if it is bigger than the paper), or download a 300 DPI PNG or an SVG with the spiral as one smooth path. 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.
Spiral
Draw
Lines
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every line and number is worked out from r = a + bθ by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw and print an Archimedean spiral
- Pick a preset, or choose what to work out and type the other three values.
- Choose one or two arms, the direction, and whether to draw the measuring ray with a tick at every turn.
- Pick the line colour and width, the unit and the paper.
- Choose Fitted to one page, or Real size for a template at the true size.
- Download the PDF and print at 100% (Actual size), or the SVG for a plotter, laser or cutter.
What you can use it for
- Coiled rope rugs, baskets, coasters and mats: the length of rope for a size
- Lettering and text-on-a-path guides
- Flat wire and cord coils
- Maths lessons: polar curves and arc length
- Pen plotter, laser and vinyl art
- Spiral drawings, mazes and crafts
The 14 presets
| Preset | Inner radius | Gap | Turns | Outer radius | Line length |
|---|---|---|---|---|---|
| Classic spiral: 8 turns, 10 mm apart | 0 mm | 10 mm | 8 | 80 mm | 2014.7 mm |
| Tight coil: 20 turns, 3 mm apart | 5 mm | 3 mm | 20 | 65 mm | 4398.8 mm |
| Rope rug: 10 mm rope, 20 turns (tiled real size) | 5 mm | 10 mm | 20 | 205 mm | 13197.6 mm |
| Cord coaster: 6 mm cord, 8 turns (real size) | 3 mm | 6 mm | 8 | 51 mm | 1358.5 mm |
| Double spiral: two arms, 4 turns | 0 mm | 20 mm | 4 | 80 mm | 2 × 1012.3 mm |
| Clockwise spiral: 6 turns | 0 mm | 12 mm | 6 | 72 mm | 1361.8 mm |
| Lettering path: 3 turns, 12 mm apart | 20 mm | 12 mm | 3 | 56 mm | 717.3 mm |
| Flat wire coil: 15 turns, 2 mm apart (real size) | 10 mm | 2 mm | 15 | 40 mm | 2356.4 mm |
| Half-inch spacing: 6 turns | 0.5 in | 0.5 in | 6 | 3.5 in | 75.48 in |
| With the measuring ray: 5 turns | 0 mm | 15 mm | 5 | 75 mm | 1183.6 mm |
| Large centre: 50 mm inner radius, 6 turns | 50 mm | 8 mm | 6 | 98 mm | 2790.2 mm |
| Many turns: 40 turns, 1.5 mm apart | 0 mm | 1.5 mm | 40 | 60 mm | 7540.6 mm |
| Double spiral, clockwise, with the ray | 0 mm | 16 mm | 5 | 80 mm | 2 × 1262.5 mm |
| Two and a half turns | 0 mm | 20 mm | 2.5 | 50 mm | 399 mm |
The outer radius is inner + gap × turns. The length is the length of the line itself, worked out from the exact formula.
Rope for a coiled rug or mat
| Rope width | Mat width | Turns | Rope length (centre line) |
|---|---|---|---|
| 6 mm | 250 mm | 20.33 | 8.18 m |
| 8 mm | 400 mm | 24.5 | 15.70 m |
| 10 mm | 600 mm | 29.5 | 28.27 m |
| 12 mm | 900 mm | 37 | 53.01 m |
Worked out with touching coils (gap = rope width), starting half a rope width from the centre. This is the geometric length only: allow extra for knots, joins and squash.
The formula
| Part | Meaning |
|---|---|
| r = a + bθ | Radius grows by b for every radian turned |
| Gap = 2πb | The same distance between neighbouring turns everywhere |
| Outer = inner + gap × turns | Any three of these give the fourth |
| Two arms | The second arm is the first turned 180°: the lines alternate half a gap apart |
What prints
| Download | What is in it |
|---|---|
| The spiral fitted to one page or at real size (tiled if needed), with the numbers and a check bar | |
| PNG 300 DPI | The same sheets as pictures |
| SVG (real size) | Only the drawing: each arm as one smooth path, the ray as a separate path |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| An evenly spaced spiral, its length and a template | Archimedean Spiral (this page) |
| A golden or Fibonacci spiral | Golden Spiral |
| A photo drawn as one spiral line | Spiral Art Generator |
| Gear-and-wheel loop curves | Spirograph Generator |
| A hanging chain or arch curve | Catenary |
Different tools: the Spiral Art Generator shades a photo along a spiral line, the Spirograph Generator draws gear curves and the Mandala Maker builds radial designs.
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Frequently asked questions
What is an Archimedean spiral?
A spiral that moves outwards by the same distance on every turn, like a coiled rope. In polar form r = a + bθ: a is the radius where it starts, and each full turn adds 2πb. It is named after Archimedes, who described it in his book On Spirals in the 3rd century BC.
How is it different from a golden or logarithmic spiral?
In a logarithmic spiral, such as the golden spiral, each turn is a fixed number of times wider than the last, so the gaps keep growing. In an Archimedean spiral the gap stays the same on every turn. Our tests check that the gap is the same at every turn to one billionth.
Which values can I type?
Any three of the inner radius, the outer radius, the number of turns and the gap between turns. The fourth follows from outer = inner + gap × turns, and the tool fills it in.
How long is the spiral line?
The tool works it out with the exact formula for the arc length, (b/2)[φ√(1 + φ²) + asinh φ] between φ = inner/b and φ = outer/b, where b = gap / 2π. Our tests check it against a numerical integration to one millionth.
How much rope do I need for a coiled rug?
Set the gap to the rope’s width (touching coils) and the outer radius to half the rug’s width. The tool’s length is the length of the spiral through the middle of the rope. Real rope squashes and stretches, so allow extra; the table below gives a few sizes worked out the same way.
What is a double spiral?
Two arms, the second turned half a turn from the first. Along any line from the centre the two arms alternate, half a gap apart, which our tests check.
How smooth is the SVG?
Each arm is one path of cubic curves. The tool adds curves until the path is within 0.01 mm of the true spiral at the printed size, and our tests check that on thousands of points.
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.
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.