Radians Versus Degrees

This page compares and contrasts the two systems of measuring angles in math: radians and degrees, and explains why radians are the preferred. This page assumes a reasonable level of familiarity with the degrees system and an introductory level of experience with radians.

Short Answer

The short answer for why radians are preferred is that the radian system leads to more succinct and elegant formulas throughout mathematics[1]. For example, the derivative of the sine function is only true when the angle is expressed in radians.

Long Answer

This figure illustrates the degree angle system which represents a full rotation as 360 units called degrees.
Figure 1: Degree Angle System
This figure illustrates the radian angle system annotated using the circle constant τ (tau).
Figure 2: Radian Angle System

There are two systems for measuring degrees in math: degrees and radians. The degree system divides a full rotation into and often introduces the concept of angles to students. The degree system is rooted in the origins of trigonometry and is a useful system for measuring angles. However, as time has gone by and math has evolved, the radian system has replaced degrees as the preferred system because of its natural connection to the geometry of the circle.

Radius Invariance

The concept of angles and circles are intimately related. Consider the angle (theta) formed by two rays whose vertex is at the center of two cocentric circles of radii , and arc-lengths , . Observe that the angle can be defined both in terms of the first circle and the second circle. This is illustrated below.

Radius Invariance Property
Figure 3: Radians (Radius Invariance Property)

This observation suggests a natural definition for radians. In both cases, the arc-length is proportional to the radius which implies that ratio of arc-length to radius is the equal. This is stated more formally in math notation as:

This relationship gives us the definition of radians: A radian angle is equivalent to the ratio of arc-length to radius, as shown in the equation below. Since radians measure ratio this is why they are referred to as a “dimensionless unit” or having a “radius-invariant” property.

Note, the equivelant symbol () represents that all angles on the domain of real numbers are equivalent to a corresponding angle in the domain of to a full rotation. For example, the angles and , measured using the circle constant , are equivalent.

Circle Constant

Definition of the circle constant τ (tau)
Figure 4: τ (tau) Definition

As a result of the definition of radians discussed above, a full rotation or “one turn” in radians is approximately equal to . This special number is called the circle constant represented by the symbol (tau) and appears in many areas of math, statistics and physics. One way to visualize this number is to divide any circle by its radius as shown below.

This figure illustrates the geoemtric definiton of a radian angle.
Figure 5: Radians Definition

The circle constant makes talking about and using radians much easier. Take, for example, the interactive below which visualizes angles measured in radians and annoted using the circle constant. Everything becomes a fraction of one “turn”.

Elegance

The claim here is that the radians system leads to succinct and elegant formulas. Here we see a relationship between the functions sine and cosine that wouldn’t be true if we had instead used the degree angle system. If we take the derivative of sine expressed below as a power series, we are left with the power series definition of cosine.

Applying the same process we can show that the derivative of cosine is equal to negative sine.

Formulas

The advantages of the radian system become apparent when definining complex formulas such as the area of the circle, definition of the trigonometric function sine and cosine, normal distribution and Euler’s formula.

This section is still in development, the derivations are hopefully coming soon. Thanks for your patience!

TODO: derive area of circle formula

TODO: derive definition of sine and cosine

TODO: derive normal distribution formula

TODO: euler’s formula

Links

References

  1. No, really, pi is wrong: The Tau Manifesto
    Michael Hartl