Relation Between Degree and Radian

The Fundamental Relation Between Degrees and Radians

The entire bridge between degree measure and radian measure rests on one simple observation: a full circle is simultaneously 360° and $2\pi$ radians. Since both measures describe the same complete rotation, they must be equal.

Dividing both sides by 2 gives the most important conversion relation you will use:

This single equation is the key that unlocks every conversion between the two systems. From it, we can derive two working formulas.

These are not separate facts — they are just the same equation $\pi = 180^\circ$ rearranged. If you ever forget them, just start from $\pi = 180^\circ$ and solve for whatever you need.

Approximate Numerical Values

Using $\pi \approx \frac{22}{7}$, we can get practical approximations:

Since $0.27^\circ = 0.27 \times 60 = 16.2$ minutes, we write:

Going the other way:

Table of Common Angles

The following table shows the most frequently used angles in both systems. Memorising these will save you enormous time.

Notice the pattern: $30^\circ$ increments give $\frac{\pi}{6}$ increments in radian measure. This is no coincidence — it follows directly from $\pi = 180^\circ$.

Notational Convention

When we write $\theta^\circ$, we mean the angle whose degree measure is $\theta$. When we write just $\beta$ (without a degree symbol), we mean the angle whose radian measure is $\beta$. …