Reuleaux Triangle
Make a Reuleaux triangle, the curved triangle that is the same width in every direction: three arcs, each centred on one corner of an equilateral triangle and drawn with the side as radius. Set the width, or choose a Reuleaux pentagon, heptagon or any odd number of sides up to 11. Show the polygon, the compass circles and a width gauge, watch the shape roll between two parallel lines without ever lifting the top one, and print it at real size, as a solid shape or a whole sheet of copies to cut out. Download a PDF, a 300 DPI PNG or an SVG drawn with exact arcs. 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.
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Runs in your browser with our own code: nothing you type is uploaded or saved. Every arc and number is worked out from the width by our own code. This is geometry only. You may use what you make here, including commercially, with no credit needed.
How to draw a Reuleaux triangle and print it at real size
- Pick a preset, or choose the number of sides and type the width.
- Tick the polygon, the corner dots, the compass circles or the width gauge (and its direction).
- Press Play to watch the shape roll between two lines, or tick the rolling diagram to print it.
- Choose Fitted to one page, Real size, or a sheet of copies to cut out; pick the paper.
- Download the PDF and print at 100% (Actual size), or the SVG for a cutter or laser.
What you can use it for
- Maths lessons: shapes of constant width and Barbier’s theorem
- Cut-out shapes for class demonstrations of rolling between two rulers
- Templates for crafts, coasters, tiles and stencils
- Gothic-style window tracery and decorative design
- Laser, vinyl and plotter cutting with exact arcs
- Comparing the Reuleaux triangle with Reuleaux polygons of more sides
The 14 presets
| Preset | Sides | Width | Perimeter | Area | Shapes per sheet |
|---|---|---|---|---|---|
| Reuleaux triangle: 50 mm wide, triangle shown | 3 | 50 mm | 157.1 mm | 1761.9 mm² | 1 |
| Reuleaux triangle: 3 in wide (real size) | 3 | 3 in | 9.42 in | 6.3429 in² | 1 |
| Sheet of 40 mm triangles to cut out | 3 | 40 mm | 125.7 mm | 1127.6 mm² | 20 on US Letter |
| Rolling between two lines: 50 mm triangle | 3 | 50 mm | 157.1 mm | 1761.9 mm² | 1 |
| Width gauge at 20°: 80 mm triangle | 3 | 80 mm | 251.3 mm | 4510.5 mm² | 1 |
| Compass construction: three circles of radius w | 3 | 60 mm | 188.5 mm | 2537.2 mm² | 1 |
| Solid shape: 100 mm triangle (real size) | 3 | 100 mm | 314.2 mm | 7047.7 mm² | 1 |
| Large triangle: 300 mm wide (tiled real size) | 3 | 300 mm | 942.5 mm | 63429 mm² | 1 |
| Reuleaux pentagon: 60 mm wide | 5 | 60 mm | 188.5 mm | 2730.6 mm² | 1 |
| Reuleaux pentagon rolling between two lines | 5 | 40 mm | 125.7 mm | 1213.6 mm² | 1 |
| Reuleaux heptagon: 60 mm wide | 7 | 60 mm | 188.5 mm | 2779 mm² | 1 |
| Reuleaux 9-gon: 60 mm wide, width gauge | 9 | 60 mm | 188.5 mm | 2798.4 mm² | 1 |
| Reuleaux 11-gon: 60 mm wide | 11 | 60 mm | 188.5 mm | 2808.1 mm² | 1 |
| Sheet of 30 mm pentagons to cut out | 5 | 30 mm | 94.2 mm | 682.65 mm² | 35 on US Letter |
Every value is worked out by the tool from the width and the number of sides. The perimeter is π × width for all of them.
Reuleaux triangle sizes
| Width | Perimeter (πw) | Area ((π − √3)w²/2) | Corner to centre |
|---|---|---|---|
| 25 mm | 78.5 mm | 440.48 mm² | 14.4 mm |
| 50 mm | 157.1 mm | 1761.9 mm² | 28.9 mm |
| 100 mm | 314.2 mm | 7047.7 mm² | 57.7 mm |
| 1 in | 3.14 in | 0.70477 in² | 0.58 in |
| 2 in | 6.28 in | 2.8191 in² | 1.15 in |
| 3 in | 9.42 in | 6.3429 in² | 1.73 in |
Reuleaux polygons compared
| Sides | Corner to centre | Area | Area as % of a circle of the same width | Perimeter |
|---|---|---|---|---|
| 3 | 0.5774 w | 0.70477 w² | 89.73% | πw = 3.14159 w |
| 5 | 0.5257 w | 0.75850 w² | 96.57% | πw = 3.14159 w |
| 7 | 0.5129 w | 0.77194 w² | 98.29% | πw = 3.14159 w |
| 9 | 0.5077 w | 0.77732 w² | 98.97% | πw = 3.14159 w |
| 11 | 0.5051 w | 0.78002 w² | 99.31% | πw = 3.14159 w |
w is the width. More sides bring the shape closer to a circle; the perimeter stays πw.
What prints
| Download | What is in it |
|---|---|
| The shape fitted to one page, at real size (tiled if needed) or as a sheet of copies, with the numbers and a check bar | |
| PNG 300 DPI | The same sheets as pictures |
| SVG (real size) | Only the drawing: each shape as one closed path of exact arcs, guides as separate parts |
For a cutter that needs DXF, use SVG to DXF; for your own cut-out lettering and shapes, the Stencil Maker.
Which tool to use
| You want | Use |
|---|---|
| A shape of constant width, its numbers and a template | Reuleaux Triangle (this page) |
| Circles in exact sizes | Circle Template |
| The two-circle lens shape | Vesica Piscis |
| A figure-eight curve with foci | Lemniscate |
| Nets to fold 3D shapes | 3D Shape Nets |
Other drawing tools: 3D Shape Nets prints folding nets, the Truchet Tiles generator makes tile patterns and the Mandala Maker builds radial designs.
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Frequently asked questions
What is a Reuleaux triangle?
A curved triangle made of three circular arcs. Draw an equilateral triangle with side w; with the compass at each corner draw the arc of radius w that joins the other two corners. The shape is w wide in every direction. It is named after Franz Reuleaux, a 19th-century German engineer.
What does constant width mean?
Put the shape between two parallel lines that both touch it: however you turn it, the lines are always the same distance apart, the width w. A circle has this property and so does every Reuleaux polygon. Our tests check the width of every shape on this page in 3,600 directions to one billionth.
Why does it roll between two lines?
Because its width never changes, the shape can turn between two parallel lines w apart and touch both all the time. The animation and the rolling diagram are drawn from the same geometry as the template: the lowest point stays on the bottom line and the highest on the top line in every position, which our tests check.
What is the area of a Reuleaux triangle?
(π − √3)w²/2, about 89.7% of a circle of the same width. For a width of 50 mm it is 1761.9 mm². Our tests check the formula against numerical integration.
How long is its perimeter?
πw, the same as a circle of the same width: each of the three arcs is a sixth of a circle of radius w. For 50 mm that is 157.08 mm. Every shape of constant width w has the perimeter πw (Barbier’s theorem); the tool shows it for every number of sides.
What are Reuleaux polygons?
The same construction on a regular polygon with an odd number of sides: each corner is the centre of an arc joining the two corners opposite it. They are also of constant width, and with more sides they look more like a circle (the table below gives the areas).
Can I draw one with a compass?
Yes. Draw a line of length w, set the compass to w and draw arcs from both ends to find the third corner. Then draw the three arcs between the corners, each from the opposite corner. Tick the compass circles to print the full circles as guides.
Is the SVG exact?
Yes. Each side is one SVG arc command of radius w between two corners, so a cutter or laser that reads arcs follows the true circle. The PDF draws each arc as short cubic curves, which our tests keep within 0.01 mm of the arc.
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.