Plotting a polar point
A polar point (r, θ) is a distance and an angle. Find the spoke for θ, starting from 0° on the right and turning counterclockwise, then count r rings out from the centre (the pole). The point (3, 60°) is on the third ring, on the 60° spoke. The same point has many names: adding 360° to the angle, as in (3, 420°), lands in the same place.
Graphing a polar equation
Make a table of r for each spoke, then plot and join the points in order of θ. With 15° spokes that is 24 values for a full turn, enough to see the shape of most curves. OpenStax’s Precalculus names five classic polar curves: cardioids, limaçons, lemniscates, rose curves and Archimedes’ spirals. Two quick checks before you start:
- Rose curves r = a cos(nθ) and r = a sin(nθ) have n petals when n is odd and 2n petals when n is even, so r = 2 cos(4θ) has 8 petals, each 2 rings long.
- Circles r = a cos θ pass through the pole, with diameter a along the polar axis.
If r is larger than the rings on the page, let each ring stand for 2 or 5 units and write the scale on the 0° spoke.
The spokes in degrees and radians
| Degrees | Radians | Degrees | Radians | Degrees | Radians |
|---|---|---|---|---|---|
| 0° | 0 | 120° | 2π/3 | 240° | 4π/3 |
| 15° | π/12 | 135° | 3π/4 | 255° | 17π/12 |
| 30° | π/6 | 150° | 5π/6 | 270° | 3π/2 |
| 45° | π/4 | 165° | 11π/12 | 285° | 19π/12 |
| 60° | π/3 | 180° | π | 300° | 5π/3 |
| 75° | 5π/12 | 195° | 13π/12 | 315° | 7π/4 |
| 90° | π/2 | 210° | 7π/6 | 330° | 11π/6 |
| 105° | 7π/12 | 225° | 5π/4 | 345° | 23π/12 |
For finer angles, choose 5° or 10° under Spoke every. For graphs in x and y instead, use the coordinate plane with numbered axes.
How this page sets up the sheet
- Polar, spokes every 15°, labelled in degrees.
- In inches: 1/4 in (6.35 mm) between rings, 4 per inch; bold ring every 4.
- In millimetres: 5 mm (0.197 in) between rings, 2 per cm; bold ring every 4.
It opens in inches in the US and other countries that use Letter paper, and in millimetres everywhere else (or in whichever unit you last used on this site). The paper size follows your browser too: Letter in the US, A4 elsewhere. Everything stays editable. Your own page settings, such as margins, are remembered for next time; the grid itself always opens as set up here. To start from a blank setup, use the graph paper generator.