Radians Versus Degrees

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

Introduction

This figure depicts how the degrees system measures angles where one full roation is 360 degrees.
Figure 1: Degree Angle System
Radian Angle System
Figure 2: Radian Angle System

There are two systems for measuring degrees in mathematic: degrees and radians. The degree system divides a full roation into and often introduces the concept of an angle to students. The degree system is rooted in the first origins of trigonometry and is a useful system for measuring angles. However, as time has gone by and math progressed, the radian system has replaced degrees as the preferred system.

The Short Answer

The short answer to 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 [1].

The longer answer explains why the “dimensionless” or “radius-invariant” property of radians leads to these more succinct and elegant formulas.

The Long Answer

The concept of angles and circles are intimately related. Consider the angle formed by two rays as illustrated in figure 3 whose vertex is at the center of two cocentric circles of radii , and arc-lengths , . Observe that the angle can be defined in terms of the first circle and the second circle.

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 an angle measured in radians: A radian angle is equivalent to the ratio of arc-length to radius. This definition of radians is expressed in the notation below.

As a result of this definition, a full rotation or “one turn” in radians is approximately equal to as shown in figure 4. This special number is called the circle constant and appears in many areas of math, statistics and physics.

The geometric Definition of Radian System.
Figure 4: Radians Definition

The Circle Constant

The circle constant (tau) makes measuring angles in radians relatively straightforward and is where radians show up in numerous applications. A good rule of thumb is that if appears in formula, there usually is some notion of the circle present. The numeric value of can be calculated by dividing the circumference of any circle by its radius as shown below.

Definition of the circle constant τ (tau)
Figure 5: τ (tau) Definition

Succinct and Elegant 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

References

  1. No, really, pi is wrong: The Tau Manifesto
    Michael Hartl
    https://tauday.com/tau-manifesto