A radian is a unit of angular measure used in mathematics and physics to express angles. It is defined as the angle subtended when you take the radius of a circle and wrap it around the circumference of the circle. One radian is the measure of a central angle that intercepts an arc equal in length to the radius.
Mathematically, if s is the arc length, r is the radius, and θ is the angle in radians, the relationship is given by the formula: θ = s / r
In a full circle, there are 2π radians because the circumference of a circle is 2πr, and if you travel the entire circumference, you've covered 2π radians.
Conversions between degrees (°) and radians can be done using the fact that 180° = π radians. To convert from degrees to radians, you multiply the number of degrees by π/180, and to convert from radians to degrees, you multiply the number of radians by 180/π.
Radians are often favored in mathematical analysis and calculus because they simplify many mathematical formulas involving trigonometric functions. They also make certain mathematical relationships more intuitive, especially when dealing with circular motion and calculus operations involving angles.