Degrees and Radians

Trigonometry · Axiom Academy

Understanding two fundamental ways to measure angles: the familiar degree system and the mathematically natural radian system. 1. Degrees: Dividing the Circle Why 360? Ancient Babylonian astronomers used a base-60 number system and noticed that 360 is divisible by many numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, 15...), making it convenient for dividing circles into equal parts. It's also close to the 365 days in a year. Common Angle Measures in Degrees Right angle: 90° (quarter turn) Straight angle: 180° (half turn) Full rotation: 360° (complete turn) Acute angle: Between 0° and 90° Obtuse angle: Between 90° and 180° 2. Radians: Arc Length and Radius This definition creates a beautiful relationship: . When we measure angles in radians, arc length calculations become remarkably simple—just multiply the angle by the radius! Since a full circle has circumference , and each radian corresponds to an arc length of , a complete circle contains radians. 3. The Key Relationship: π Radians = 180° Since a full circle is both 360° and radians, we can establish the fundamental conversion: 4. Converting Between Degrees and Radians Using the relationship , we can derive conversion formulas: Convert π/3 radians to degrees:

This is the written version of the interactive lesson above. See the full Trigonometry course.