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Explore special angles around the unit circle

By Published
  • Learn
  • Unit circle

Follow short visual explanations from your first radian to special angles around the full circle. Connect the patterns, control the pace, and practice matching degrees with radians inside each lesson.

Learn progress and Angle Match rounds are not saved to your account yet. Lessons are available to approved members.

At a glance

  • Build from radians to a full circleStart with first-quarter special angles, reflect into the second quadrant, then connect the remaining angles and axis landmarks.
  • Follow the explanation at your pacePause and replay the demonstrations, adjust playback from 0.5× to 2×, or open the lesson text to read along.
  • Put the angle pairs into practiceMatch degrees and radians inside each lesson, with an optional whole-circle review in Full Circle.

The Unit Circle lessons bring explanations, visual demonstrations and practice together in one place. This is an introduction to the sequence already available in Learn, so you can see what it covers and choose where to begin.

Start with the meaning of radians

Special Angles begins with a circle of radius 1 and connects the length of an arc to a turn. From there, it builds the familiar first-quarter pairs: 30° and π/6, 45° and π/4, 60° and π/3, and 90° and π/2.

The demonstrations show how dividing a half turn creates these angles. A separate pattern step brings together the denominators 6, 4, 3 and 2. As the denominator gets smaller, the fraction of π and the angle get larger. The aim is to connect a memory cue with a picture that explains it.

Carry the familiar positions around the circle

Second Quadrant reflects those positions across the vertical axis. It shows how each angle sits one reference-angle piece short of a half turn, helping explain values such as 5π/6.

Full Circle continues into the lower half, connects the axis landmarks, including 270° and 3π/2, and brings the special angles into one view. Its patterns help you find the number before π while keeping the familiar reference-angle denominators.

These lessons focus on angle positions and degree–radian pairs. They do not yet teach the complete set of sine and cosine values.

Read, watch and practice at your pace

Each demonstration starts paused. Play it when you are ready, pause on a step you want to inspect, or replay the explanation. Text appears with the animation, and the book control opens the lesson text when you would rather read through it. Playback speed ranges from 0.5× to 2×.

Angle Match inside each lesson lets you practice choosing the radian measure for a displayed degree value. The Full Circle lesson starts with the new lower-half angles and offers a whole-circle round when you want a broader review. The separate Games activity currently covers the first-quarter special angles.

Open the lessons

Begin with Radians and special angles, then use the lesson menu to continue into Second Quadrant and Full Circle. You can return to an explanation or pattern whenever you need it.

Learn progress and Angle Match rounds are not saved to your account yet. Flashcard and quiz activity in Practice has its own tracking. Lesson access requires an approved account; see UTA Student Access for the current invitation offer.