A radian measures angles using arc length. One radian is the angle subtended when the arc length equals the radius. A full circle is 2π radians, so 180 degrees equals π radians. Radians are dimensionless, defined as a ratio of lengths.

Radians simplify many formulas. The derivative of sin x is cos x only when x is in radians. The arc length formula is s = rθ, and sector area is ½r²θ. In degrees, these formulas carry awkward conversion factors.

Calculus, physics, and engineering use radians almost exclusively. Trigonometric functions in programming languages expect radians. Converting between degrees and radians is a basic skill: multiply by π/180 to go from degrees to radians. The unit is natural for rotational motion and wave analysis.

• Angle measure based on arc length
• Full circle = 2π radians
• 180 degrees = π radians
• Preferred in calculus and physics