9.1 Arc Length and Sector Area

📐 Part 1: Radian Measure

What is a Radian?

You already know how to measure angles in degrees — a full turn is 360°. But there is another way to measure angles called radians, and this is the unit used in most advanced mathematics.

Here is the key idea:

In other words: if you take a piece of string equal to the radius of a circle and lay it along the curved edge, the angle it "opens up" at the centre is exactly 1 radian.

We write 1 radian as 1 rad or 1° (with a small raised c).

The Link Between Radians and Degrees

The circumference (the full distance around a circle) is $2\pi r$.

Since the radius is $r$, the full circumference is $2\pi$ lots of the radius length. This means the full 360° angle at the centre equals $2\pi$ radians.

Dividing both sides by 2:

This is the most important conversion fact. Memorise it.

Converting Between Degrees and Radians

There are two simple rules:

Think of it this way: to go from degrees to radians, you're replacing 180 with $\pi$, so you multiply by $\frac{\pi}{180}$.

Worked Example 1 — Converting Angles

a) Convert 60° to radians (give your answer in terms of $\pi$)

b) Convert $\dfrac{3\pi}{5}$ radians to degrees

$\frac{3\pi}{5} \times \frac{180}{\pi} = \frac{3 \times 180}{5} = \frac{540}{5} = 108°$