# Radian Angle System Figure 1: Radian Angle System

Radians are a unit that measure 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.

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 the 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.

### Definition Figure 2: Radians Definition

Radians are a unit that measure angle as the ratio of the angle’s arc-length over the radius of a circle. The (equivalent) symbol is used to represent the radius-invariant property of the definiton. A full rotation is equal to (tau) radians where is the naturally occuring circle constant defined by a circle’s circumference 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. Figure 3: 1/12 τ Radians Figure 4: 1/8 τ Radians Figure 5: 1/6 τ Radians Figure 6: 1/4 τ Radians

### Trigonometric Functions

The trigonometric functions and the unit circle are often where radians are introduced. Shown below are the plots of sine and cosine labeled in radians. Figure 7: Cosine Plot Figure 8: Sine Plot

## Radians vs Degrees Figure 9: Radian Angle System Figure 10: Degree Angle System

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

See the long answer on this page.