Angles and Radian Measure

The radian is a unit of plane angle, equal to 180/π (or 360/(2π)) degrees, or about 57.2958 degrees (approximately 57°17'45").It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level.

The radian is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as " 1.2 rad " or " 1.2^c " (the second symbol is often mistaken for a degree: " 1.2° "). However, the radian is mathematically considered a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ⁰ is used.

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ.

It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.

An angle is determined by rotating a ray about its endpoint. The starting position of the ray is called the initial side of the angle. The ending position of the ray is called the terminal side. The endpoint of the ray is called its vertex. Positive angles are generated by counter-clockwise rotation. Negative angles are generated by clockwise rotation. Consequently an angle has four parts: its vertex, its initial side, its terminal side, and its rotation.

An angle is said to be in standard position when it is drawn in a Cartesian coordinate system in such a way that its vertex is at the origin and its initial side is the positive x-axis.

One way to measure angles is in radians. To signify that a given angle is in radians, a superscript c, or the abbreviation rad might be used. If no unit is given on an angle measure, the angle is assumed to be in radians.