Cardioid
Draw a cardioid, the heart-shaped curve r = 2a(1 − cos θ), three ways that all give the same curve: a circle rolling round an equal circle, a family of circles through one point, and the times-table chords (join point n to point 2n round a circle). Start from one of 14 presets, including a numbered worksheet to join by hand, watch the circle roll, and download a PDF at real size on US Letter or A4, a 300 DPI PNG or an SVG for a pen plotter or laser. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). The SVG holds the drawing at the size you typed: the curve as one line, each chord as its own line, for a pen plotter, laser or cutter.
Cardioid
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Runs in your browser with our own code: nothing you type is uploaded or saved. Every line is worked out from the cardioid formulas by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw and print a cardioid
- Pick a preset, or tick what to draw: the curve, the rolling circle, circles through the cusp or times-table chords.
- Set the number of circles or points (the times table joins point n to point 2n).
- For a class worksheet, tick Number the points and Leave the chords for me to draw.
- Pick the colours, line widths, size and paper.
- Download the PDF and print at 100% (Actual size), or the SVG for a plotter or cutter.
What you can use it for
- Maths lessons: polar curves, envelopes and modular arithmetic
- Times-table circle worksheets for a class
- Pen plotter, laser engraving and vinyl art
- Posters, cards and wall art
- Thread and nail art outlines (the chord pattern)
- Sound lessons: the shape behind the cardioid microphone pickup pattern
The 14 presets
| Preset | What is drawn | Turn |
|---|---|---|
| Rolling circle: one circle round an equal circle | curve, rolling circle | cusp on the right |
| The curve on its own | curve | cusp on the right |
| Curve, cusp on top | curve | cusp on top |
| Envelope of 12 circles, with the curve | curve, 11 circles | cusp on the right |
| Envelope of 24 circles | 23 circles | cusp on the right |
| Envelope of 48 circles | 47 circles | cusp on the right |
| Times table ×2: 60 points | 59 chords on 60 points | cusp on the right |
| Times table ×2: 100 points | 99 chords on 100 points | cusp on the right |
| Times table ×2: 200 points | 199 chords on 200 points | cusp on the right |
| Times table ×2: 360 points | 359 chords on 360 points | cusp on the right |
| Times table ×2: 200 points, with the curve | curve, 199 chords on 200 points | cusp on the right |
| Worksheet: 36 numbered points to join | 36 numbered points (no chords) | cusp on the right |
| Worksheet answer: 36 points joined | 35 chords on 36 points | cusp on the right |
| All three ways on one drawing | curve, rolling circle, 23 circles, 99 chords on 100 points | cusp on the right |
The times-table presets join point n to point 2n. The worksheet preset prints the numbered points only, so a class can draw the chords by hand and see the cardioid appear.
Three ways to draw it
| Construction | How it works | Where it touches the cardioid |
|---|---|---|
| Rolling circle | A circle of radius a rolls round a fixed circle of radius a; a pen on its rim draws the curve | The pen point itself |
| Envelope of circles | Circles with centres on a circle of radius a, all passing through the cusp | At the point of the curve in the direction of the circle centre |
| Times-table chords | N points on a circle of radius 3a; join point n to point 2n | A third of the way along each chord |
All three are drawn in the same frame, so you can lay one over another (the All three ways preset). Our tests check each touch point and that each circle or chord meets the curve without crossing it.
Size, length and area
| a (rolling circle radius) | Length 16a | Area 6πa² | Chord through the cusp 4a | Curve width × height |
|---|---|---|---|---|
| 25.4 mm | 406.4 mm | 12161 mm² | 101.6 mm | 114.3 × 132.0 mm |
| 30 mm | 480.0 mm | 16965 mm² | 120.0 mm | 135.0 × 155.9 mm |
| 50 mm | 800.0 mm | 47124 mm² | 200.0 mm | 225.0 × 259.8 mm |
The PDF prints a, the length and the area for the size you choose, and a 1 in (or 2 cm) bar to check the print scale.
What prints
| Download | What is in it |
|---|---|
| The page at real size, vector: the drawing, the numbers for your size and a check bar | |
| PNG 300 DPI | The same page as a picture |
| SVG (real size) | Only the drawing: the curve as one path, each chord and circle as its own element |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| A cardioid, by rolling circle, circles or chords | Cardioid (this page) |
| Curves with more cusps (nephroid and others) | Epicycloid |
| Gear-and-wheel curves with a pen hole inside the wheel | Spirograph Generator |
| A photo turned into thread lines on nails | String Art Generator |
| A blank polar grid | Polar Graph Paper |
Different tools: the Spirograph Generator draws gear curves with the pen inside the wheel (no cusps), the String Art Generator plans a thread path from a photo, and the Guilloche Pattern Generator draws banknote-style line patterns.
More free tools
Curves, patterns and generative art?
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Frequently asked questions
What is a cardioid?
A heart-shaped curve with one sharp point, the cusp. It is the path of a point on the rim of a circle rolling round the outside of another circle of the same size. About the cusp its polar equation is r = 2a(1 − cos θ), where a is the radius of the circles.
Where does the name come from?
From the Greek kardia, heart. Giovanni Salvemini coined the name cardioid in 1741.
How do the times-table chords make a cardioid?
Put N evenly spaced points on a circle and number them 0 to N − 1. Join each point n to point 2n, taking away N when 2n is N or more. Every chord touches the same curve, and that curve is a cardioid. The tool checks every chord: it touches the cardioid at exactly one point, a third of the way along it.
How do circles make a cardioid?
Take a circle and one point on it (the cusp). Draw circles whose centres lie on the first circle and which all pass through that point. Their outline (the envelope) is a cardioid, and our tests check that every circle touches it.
Are the three drawings really the same curve?
Yes. With the rolling circles of radius a, the circles drawn round a circle of radius a and the chords on a circle of radius 3a, all three give the same cardioid in the same place. Our tests check every touch point against r = 2a(1 − cos θ) to one billionth.
What are the area and the length of a cardioid?
The area is 6πa² and the whole curve is 16a long, where a is the radius of the rolling circle. The tool shows both for your size, and our tests check them against a numerical integration.
How wide and tall is it?
4.5a from the cusp side to the far side and 3√3·a (about 5.2a) across. Every chord through the cusp is 4a long.
Where else does the cardioid turn up?
The main body of the Mandelbrot set is bounded by a cardioid, light reflected inside a round cup can make a cardioid-shaped bright line, and cardioid microphones are named after the shape of their pickup pattern.
Can I use it on a pen plotter, laser or cutter?
Yes. The SVG download is the drawing at the real size you typed: the curve is one unbroken path and every chord or circle is its own element, so a plotter draws each in one stroke. 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 is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.