Degrees and Radians: Measuring Angles Two Ways
1. Two Ways to Measure Angles
Degree Measure — A Gift from Babylon
The degree is the older of the two systems. Ancient Babylonian astronomers divided the sun's yearly path across the sky into 360 equal parts, because they estimated the year at roughly 360 days — each day the sun moved about one step along its circular journey. The Greeks adopted this convention because 360 is a highly composite number: it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180, making it easy to split a circle into many equal parts.
So one degree (°) is defined as $\frac{1}{360}$ of a full rotation.
Radian Measure — The Natural Choice
The radian is defined geometrically: one radian is the central angle that intercepts an arc whose length equals the radius of the circle. Because it is defined as a ratio of two lengths ($\theta = \frac{s}{r}$), the radian is dimensionless — it carries no physical unit. It is the SI derived unit for plane angle.
University-level mathematics works almost exclusively in radians because they make formulas in calculus, differential equations, and physics dramatically simpler (we will see why in Section 7).