Radians

Degrees are convenient, but they're arbitrary — someone chose 360 for a full circle. Radians are the natural unit: they come directly from the circle's own geometry, and they make calculus formulas clean.

What one radian is

Take the radius of a circle. Lay it along the arc. The angle at the center subtended by that arc is 1 radian.

Since the full circumference is 2πr, you can fit exactly 2π radii around the circle. So a full turn = 2π radians. That's about 6.28 radii — as shown by the three colored arcs above.

The key conversions

Starting from 2π rad = 360°:

To convert: degrees × π/180 = radians. Radians × 180/π = degrees.

Why radians make calculus work

In degrees, the derivative of sin is (π/180) cos — an ugly constant appears. In radians, the derivative of sin is simply cos. The constant vanishes because radians measure arc length directly: when θ is in radians, the arc length along the unit circle is exactly θ.

In practice: use degrees when talking to people, use radians when computing or writing formulas. Most calculators have both modes — always check which one is active.