Radians measure angles as the ratio of the angle’s arc length over the radius of the circle used to draw the arc. A full rotation in radians is equal to (tau) radians.

Note: This website uses the constant (tau) instead of (pi) as the default circle constant. The substitution can be used to translate between the two constants.

## Usage

The radian system is used for measuring angles and as the unit of choice for trigonometric functions. While the degree angle system is often used to introduce concepts, the radian system eventually becomes the preferred unit for measuring angles in math[1].

### Definition

Radians are a unit that measures angle as the ratio of the angle’s arc length over the radius of a circle. A full rotation in radians is equal to (tau) radians, where tau is defined as the circumference of a circle divided by its radius.

### Measuring Angles

Angles measured using radians are usually expressed using the circle constant (tau). Shown below are some examples of angles measured using radians. The variable (theta) is a variable commonly used for angles. By convention, angles in the coordinate plane are measured from the positive direction where the counter-clockwise rotation is positive.

### Trigonometric Functions

Radians are often introduced when the trigonometric functions are defined using the unit circle. Shown below are the graphs of the sine and cosine function labeled in radians.

The short answer for why radians are preferred to degrees is that the radian system leads to more succinct and elegant formulas throughout mathematics[2]. For example, the derivative of the sine function is only true when the angle is expressed in radians. Read more…