Blank Unit Circle

§1What each version contains

• Blank: the circle, its axes and the 16 special points, with a line at every point for the degrees, the radians and the coordinates. This is the classic blank unit circle quiz.
• Degrees only, radians only and coordinates only: one ring is printed and the others are blank. Use them to practice one connection at a time, such as radians to coordinates.
• Filled in: the complete unit circle chart with degrees, radians and (cos θ, sin θ) at every special angle. It makes a good reference sheet for a binder or a classroom wall.
• Worksheet + answer key: twelve angles in a table, each given in degrees or in radians, with the other columns to fill in exactly: radians or degrees, cos θ, sin θ and tan θ. Page 2 is the answer key. “New worksheet” draws a different set.

§2How to fill in a blank unit circle

You don’t need to memorize 48 separate labels. Fill the circle in this order and most of it writes itself:

1. The axes. 0°, 90°, 180° and 270° are 0, π/2, π and 3π/2, at (1, 0), (0, 1), (−1, 0) and (0, −1).
2. The 45° family. 45°, 135°, 225° and 315° are π/4, 3π/4, 5π/4 and 7π/4. Every coordinate is ±√2/2.
3. The 30° family. 30°, 150°, 210° and 330° all have denominator 6 in radians (π/6, 5π/6, 7π/6, 11π/6). The point is wide and low: (±√3/2, ±1/2).
4. The 60° family. 60°, 120°, 240° and 300° all have denominator 3 (π/3, 2π/3, 4π/3, 5π/3). The point is narrow and tall: (±1/2, ±√3/2).
5. The signs. x is negative on the left half and y is negative on the bottom half.

Check your work against the filled-in version, or against the interactive unit circle.

§3Using it to memorize the circle

Filling in a blank circle from memory is one of the most effective ways to learn it. Print a few copies and fill one in each day, timing yourself, and watch the time drop as the patterns stick. Between paper sessions, the fill-the-circle game in unit circle practice checks each label as you drop it, so mistakes don’t stick.

§4Printing tips

• Letter (8.5 × 11 in) is the default in the US, Canada and a few other countries; A4 is used almost everywhere else. The paper setting is remembered on this device.
• Print at 100% (“Actual size”) so the circle fills the page as designed. “Fit to page” also works but adds a margin.
• The printout contains only the page: no menus, no navigation and no ads.
• Download PDF saves the same page. The blank and filled-in versions in Letter size are also available directly as PDF files.

§5For teachers

You may print and copy these sheets for your own classes, including handing them out to students. Reselling them is not allowed; see the terms of use. The worksheet number is printed on the page, so you can generate several versions for the same class and match each one to its answer key.

§6FAQ