Radians

Radian Angle System
Figure 1: Radian Angle System

Radians are a unit that measure angles. A full rotation in radians is equal to (tau) radians, where the constant is approximately equal to . Radians are the preferred unit for measuring angles, as opposed to degrees, since they lead to “more succinct and elegant formulas throughout mathematics"[1].

Definition

The geometric Definition of Radian System.
Figure 2: Radians Definition

A radian angle is equivalent to the ratio of the arc-length over the radius of the circle. By design, the unit is “dimensionless” or, in plain language, the length of the radius is arbitrary; one radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle. The symbol (equivalent) is used to indicate that negative angles or angles more than a full rotation are mapped to the equivalent angle between and a full rotation.

This “radius-invariant” property plays an important role for why radians are the preferred unit for measuring angles. See radians vs. degrees.

Usage

The radian system is used in many places in mathematics: measuring angles, numerous geometric and advanced math formulas via the circle constant, and as the unit of choice for the trigonometric functions.

Circle Constant

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

The circle contanstant (tau) makes talking about and using the radians angle system much more convenient. The circle constant is approximately equal to [2] as illustrated by figure 3.

Definition of the circle constant τ (tau)
Figure 3: τ (tau) Definition

Without the circle constant, angles measured in radians can be confusing. Take for example, the figures figure 4 and figure 5 showing measured angles of 1 radians and 2 radians. Both figures draw tic marks at the numeric values of radians, which conceptually demonstrates what a radian is. However, as we will later see it is much more convenient to use a constant.

Angle measured as 1 radians.
Figure 4: 1 Radian
An angle measured as 2 radians.
Figure 5: 2 Radians

Angles

Angles measured using radians are expressed as a fraction of “one turn” where one turn is equal to the circle constant (tau). By convention, angles in the XY coordinate plane are measured from the positive direction where the counter-clockwise rotation is positive.

This is where the circle constant plays a crucial role in making the radians system complete; angles can easily be expressed as fractions of the whole. As figure 6 and figure 7 illustrate, the circle can be divided into equal parts and using the circle constant constant leads to elegant and succinct notation.

1 over 4 Rotation Radians
Figure 6: 1 over 4 Rotation Radians
1 over 6 Rotation Radians
Figure 7: 1 over 6 Rotation Radians

As a final note on measuring angles, here are some “special angles” that show up in numerous places. The figure also illustrates why circle constant is also the best method for converting between radians and degrees[3]

Radians Unit System Fractions τ (tau)
Figure 8: Radians Special Angles

Common Angles

Shown below are some special angles measured using radians. Note, the symbol (theta) is often used to represent a measured angle.

This figures illustrates the standardized angle that represents an twelfth turn.
Figure 9: 1/12 τ (tau) Radians
This figures illustrates the standardized angle that represents an eighth turn.
Figure 10: 1/8 τ (tau) Radians
This figures illustrates the standardized angle that represents an sixth turn.
Figure 11: 1/6 τ (tau) Radians
This figures illustrates the standardized angle that represents an fourth turn.
Figure 12: 1/4 τ (tau) Radians

Formulas

Radians appear in numerous geometric formulas and advanced formulas in mathematics via the circle constant or or alternatively through the constant . Shown below are some common geometric formulas. Note, very likely you have seen the constant used insted of This website primarily uses instead of in equations and formulas as it is the better constant[4]. The relation can be used to translate between the two constants.

Area of Circle
Area of Circle | Formula

The area of a circle is give by one-half multiplied by τ (tau) mutliplied by the radius of the circle squared.

Circumference of Circle
Circumference of Circle | Formula

The circumference of a circle is given the constant τ (tau) multpilied by the radius of the circle, where τ = 2π.

Volume of Sphere
Volume of Sphere | Formula

The volume of a sphere is given by two-thirds multiplied by the circle constant τ (tau) multiplied by the radius cubed.

The circle constant also appears in some suprising places such as the definition of the normal distribution. When is present, there usually is some connection to circles or a circular way of thinking.

Functions

The trigonometric functions and the unit circle are often where radians are introduced. Some programming languages and spreadsheet applications only offer the trigonometric functions implemented using the radian system. This is discussed in more detail on the individual function pages, but shown below are two figures which graph the two functions using the radian system.

Plot of Cosine from 0 to τ (tau) radians.
Figure 13: Cosine Function
Plot of Sine Function from 0 to τ (tau) radians.
Figure 14: Sine Function

Radians vs Degrees

Radian Angle System
Figure 15: Radian Angle System
This figure depicts how the degrees system measures angles where one full rotation is 360 degrees.
Figure 16: Degree Angle System

There are two systems for measuring degrees in math: degrees and radians. The short answer for why radians are preferred is that the radian system leads to more succinct and elegant formulas throughout mathematics [4]. For example, the derivative of the sine function is only true when the angle is expressed in radians.

See the long answer for why radians are preferred on this page.

Convert to Degrees

To convert an angle from radians to degrees multiply by the ratio of over radians [3].

references

  1. Radians Versus Degrees
    Wumbo (internal)
  2. Approximate the Circle Constant
    Wumbo (internal)
  3. Convert Angle from Radians to Degrees
    Wumbo (internal)
  4. No, really, pi is wrong: The Tau Manifesto
    Michael Hartl