Polar Graph Paper
Polar coordinate paper with circles at an even spacing and rays every 1° to 90°, with heavier rays and degree labels where you want them, as a full circle or a half circle. Download a vector PDF that prints the rings at exactly the spacing you set.
Preview and download
Scaled-down preview; the printout is full size. Dashed lines show the margins and don’t print.
Print at 100% or Actual size, not “Fit to page”. Then check the ruler at the foot of the sheet: it should measure exactly 4 in.
- Sheet
- Letter, 8.5 × 11 in, portrait
- Pattern
- 14 rings, rays 15° apart
- Ruler
- 4 in, in the bottom margin
- 1 page, vector, no watermark
Paper
Rings and rays
Lines and colour
Margins
From each paper edge to the pattern. Most printers can’t print the outer 0.2 in or so of a sheet.
Header and mixed pages
Pages and printing
How to make polar graph paper
- Ring spacing. The distance between neighbouring circles, from 1 mm up. "Heavy ring every" makes every 4th ring, say, darker so radii are quick to count.
- Rays. "Ray every" sets the angle between rays, from 1° to 90°. Heavy rays (every 90° by default) mark the axes, and degree labels can go on every ray or every few.
- Full or half circle. A half circle uses the upper half and its diameter; on a landscape page it gets much bigger.
- Page and print. The circle is centred and sized to fit inside the margins with its labels. Print at 100% so the rings are the spacing you chose.
Choosing the angle between rays
| Ray every | Rays in a circle | In radians | Good for |
|---|---|---|---|
| 1° | 360 | π/180 | Fine plotting, protractor practice |
| 5° | 72 | π/36 | Detailed plots and antenna patterns |
| 10° | 36 | π/18 | Compass roses and wind diagrams |
| 15° | 24 | π/12 | Most textbook polar equations |
| 30° | 12 | π/6 | Unit-circle work, clock hours |
| 45° | 8 | π/4 | Quick sketches, eight-way symmetry |
| 90° | 4 | π/2 | Quadrants only |
Also available: 2°, 3°, 4°, 6°, 9°, 12°, 18°, 20° and 60°. Every step divides 180°, so full and half circles both close evenly.
Worked example: a cardioid
To plot r = 3 + 3 cos θ, use 1/2 in rings (7 fit on Letter) with one ring per unit and rays every 30°. Work out r on each ray, count that many rings out, mark the point, then join the points with a smooth curve:
| θ | 0° | 30° | 60° | 90° | 120° | 150° | 180° |
|---|---|---|---|---|---|---|---|
| r | 6 | 5.6 | 4.5 | 3 | 1.5 | 0.4 | 0 |
The curve is symmetrical about the 0° ray, so the values for 210° to 330° repeat those for 150° down to 30°. The result is a heart shape with its point at the centre, the same cardioid that describes a cardioid microphone's pickup pattern.
What polar paper is used for
- Maths. Polar equations (roses such as r = cos 3θ, cardioids, limaçons and the Archimedean spiral r = aθ, which crosses each ray at evenly spaced distances), complex numbers in polar form and the unit circle.
- Physics and engineering. Antenna radiation patterns and microphone polar patterns, where each ring often stands for a fixed step in decibels rather than a linear amount; vectors and forces; and cam profiles laid out as radius against angle.
- Navigation and weather. Compass roses, bearings and wind roses, with the labels turned off and written in clockwise from north.
- Design and craft. Mandalas and radial art; clock faces (rays every 6° with heavy rays every 30° give 60 minute marks and 12 hour marks); a 20-blade Dresden Plate quilt block (18° rays); and pie charts, where 18° rays mark 5% slices.
Printing it
Print the PDF at Actual size, then check the calibration bar at the foot of the page or measure the circle: on the starting Letter sheet the outer ring is 7 in across. A circle that comes out slightly oval means the printer is scaling the two directions a little differently, usually because of the paper feed; the guide to printing at actual size explains why that happens and what to do.
Frequently asked questions
How do I plot a point on polar graph paper?
A point (r, θ) is r units out from the centre along the ray at angle θ. Find the ray first (0° points right, angles increase counter-clockwise), then count r rings out along it. If r is negative, count the same number of rings the opposite way, along the ray at θ + 180°. Decide before you start how much one ring stands for; on paper with 7 rings, a function that reaches r = 6 fits at one unit per ring.
Which way are the angles measured?
As in mathematics: 0° points to the right (3 o'clock) and angles increase counter-clockwise, so 90° is straight up. The degree labels follow that convention. Compass bearings work differently, as the next answer explains.
Can I use polar paper for compass bearings and navigation?
Yes, but bearings are measured clockwise from north (the top of the page), so the printed labels would be wrong. Turn Degree labels off and write your own: 000° at the top, 090° at the right, 180° at the bottom and 270° at the left. Rays every 10° with heavier rays every 30° or 90° make a good compass rose.
How do I work in radians?
Pick a ray step that is a simple fraction of π: rays every 15° are π/12 apart, every 30° π/6 and every 45° π/4. With 15° rays, every second ray is a multiple of π/6 and every third a multiple of π/4, which covers the angles used in most textbook problems. Turn the degree labels off and write the radian values in by hand.
Why don’t all the rays reach the centre?
Near the centre the rays crowd together, and with 1° or 5° steps they would fill it with ink. Light rays therefore start at the first ring where neighbouring rays are at least 4 pt (about 1.4 mm) apart. Heavy rays, every 90° by default, always run all the way in, so the axes stay complete.
How many rings fit on a page?
On a portrait Letter sheet with 1/2 in margins, 14 rings of 1/4 in make a circle 7 in across, leaving room for the degree labels. On A4, 17 rings of 5 mm fit. For a half circle, landscape is the better shape: it fits 19 rings of 1/4 in on Letter.