Easy as 𝜋

I realise that my last few posts are a bit more advanced than my other posts. So let’s take it down a few notches. We will still be talking about angles however.

In my posts on trigonometry, I used angles measured in degrees. But that is a rather arbitrary measure. The origin of measuring angles based on 360° for a full circle is obscure, but one theory has it that it is based on ancient calendars that had 360 days in a year. In science and engineering, a different measure of angles is used, radians.

The number 𝜋 crops up a lot, especially when talking about circles. This is because 𝜋 is the ratio of a circle’s circumference (perimeter) to its diameter:

\[
\mathit{\pi}\hspace{0.33em}{=}\hspace{0.33em}\frac{C}{d}
\]

where C is circumference and d is diameter. This holds true no matter if the circle is as small as “o” or as large as the orbit of earth around the sun. The symbol “𝜋” is a Greek letter with the name “pi”, and that is the symbol used to represent this ratio. In fact, it is the only way to write down this number exactly if we all agree what 𝜋 represents. This is because 𝜋 is an irrational number which means you cannot ever write it down exactly with numbers. Approximately, 𝜋 is 3.14159265359… where the decimal part goes on forever without ever a repeating pattern. Irrational or not, 𝜋 is a natural number to associate with circles.

Now if you replace the diameter d with twice the radius, that is 2r, the equation becomes

\[
\mathit{\pi}\hspace{0.33em}{=}\hspace{0.33em}\frac{C}{2r}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{2}\mathit{\pi}\hspace{0.33em}{=}\hspace{0.33em}\frac{C}{r}
\]

which is again true for any size circle. Now notice that the 2𝜋 on the left side is unitless. What do I mean by that? The right side of the equation is a circumference measured say, in meters, divided by a radius, again measured in meters. If you remember my post on units, you’ll remember that units can be cancelled in a fraction just like variables. So in this case, we have meters divided by meters which cancel each other and we are left with no units. So a full circle is associated with the unitless number 2𝜋. What about a half circle? A half circle will have half the circumference. Half of 2𝜋 is 𝜋. What about a quarter circle? A quarter of 2𝜋 is 𝜋/2. As these numbers are all unitless, they were seen as a natural way to define angles.

So a half circle (also called a semi-circle), can be seen as an arc formed by the familiar angle 180°. The quarter circle is an arc formed by the angle 90°. But instead of degrees, we can call these angles 𝜋 and 𝜋/2 since these are the length of the arc formed by the angle divided by the radius. The term used to distinguish this measure of angles from degrees is radians.

This can be generalised for any angle. Please look at the below diagram:

So an angle in radians is the arclength formed by the angle divided by the radius. By the way, radians are abbreviated as rad and this is what you will probably see on your calculator when you use the radians mode.

So there are 2𝜋 radians in a full circle, 𝜋 radians in a semi-circle, and 𝜋/2 for a quarter circle. We can use the fact that 180° = 𝜋 radians to convert between the two measurements. To change degrees to radians, multiply by 𝜋/180. To change radians to degrees, multiply by 180/𝜋.

For example, 30° is 30 × 𝜋/180 = 𝜋/6 rad. It is customary to keep 𝜋 as 𝜋 when working with radians.

One advantage of using radians becomes immediately apparent when rearranging the equation in the figure above. The arclength formed by an angle measured in radians is simply r𝜃.

So now that you know what radians are, my future posts will use this measure almost exclusively.