Polar Graph Paper
Polar graph paper is a set of equally spaced rings crossed by lines from the centre at equal angles. Choose how many rings and how many degrees between the lines (30, 15, 10 or 5), with or without degree labels, and the sheet is drawn to fill your paper.
Your sheet
Printing: choose Actual size or 100%, not “Fit to page”, so 1 cm prints as 1 cm. How to check.
How is polar paper drawn?
Ring radius = outer radius × k ÷ rings, for k = 1 to rings. Lines = 360 ÷ degrees, each from the centre to the outer ring at angle k × degrees, measured anticlockwise from the right.
Worked example. Ten rings and 15° on A4 portrait: the outer radius is 91 mm (half the page width minus 14 mm for labels), so the rings are 9.1 mm apart. 360 ÷ 15 = 24 lines, labelled 0°, 15°, 30° … 345°. The 90° line points straight up, the same way angles are measured in maths.
Plotting a polar point
For (r, θ) = (3, 60°) on a ten-ring sheet where the outer ring is 10: find the 60° line, count three rings out from the centre and mark the crossing. Negative r goes the opposite way along the same line. Join the points of a curve in order of θ.
Common polar curves to try
r = 2 cos θ is a circle through the origin. r = 1 + cos θ is a cardioid. r = cos 3θ is a three-petal rose and r = cos 2θ a four-petal one. r = θ (in radians) is the Archimedean spiral. Each needs points every 15° or less to look smooth, which is why 15° is the default.
Beyond maths
Sound engineers plot microphone pickup patterns on polar paper, radio engineers plot antenna gain, and meteorologists draw wind roses with the compass directions on the lines. For a compass sheet, choose 30° lines (12 points) or 10° and label as you need.
Questions about polar graph paper
What is polar graph paper for?
Plotting points given as a distance and an angle (r, θ) rather than x and y: polar curves such as roses and spirals in maths, antenna and microphone patterns in engineering, wind roses in geography, and circular designs in art.
How many degrees between lines should I choose?
Every 15° (24 lines) is the usual classroom setting and matches the common angles π/12 apart. Every 10° suits engineering plots. Every 5° (72 lines) is for precise work and prints best on A3 or in landscape.
How many rings should I use?
Ten is the default and each ring is then one tenth of the radius, which is easy to read. Use 5 for a bold sheet, or 12 and 24 when the function’s values divide that way. Every fifth ring is drawn heavier to help counting.
Can I turn off the degree labels?
Yes, untick “Degree labels”. The labels sit just outside the outer ring at every line (every 10° when the lines are 5° apart, to stay readable).
Are the labels in radians too?
The labels are in degrees. For radians, remember 90° = π/2, 180° = π and 360° = 2π; the unit circle chart lists every common angle both ways.