1.4 Circular Measure

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand what a radian is, and convert between radians and degrees in both directions.
  2. Use the formulas s = rθ and A = ½r²θ to find arc lengths and sector areas, and apply these in problems that also involve triangles (finding lengths, angles, and areas).

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Part 1: Radians — A New Way to Measure Angles

What is a Radian?

Up until now, you have measured angles in degrees — where a full turn is 360°. But in advanced mathematics, we use a different unit called the radian.

Here is the key idea:

One radian is the angle at the centre of a circle when the arc (the curved part) in front of that angle is exactly equal in length to the radius of the circle.

Let's make this clearer with a picture in your mind:

  • Draw a circle with centre O and radius r.
  • Now mark two points, A and B, on the edge (circumference) of the circle.
  • The curved line from A to B is called an arc.
  • If the length of arc AB = r (same as the radius), then the angle AOB = 1 radian.

So a radian is defined purely from the circle itself — it is a natural unit of angle.

The Relationship Between Radians and Degrees

We know that the full circumference of a circle = 2πr. Since the radius is r, and one radian corresponds to an arc length of r, the number of radians in a full turn is:

Full turn=2πrr=2π radians\text{Full turn} = \frac{2\pi r}{r} = 2\pi \text{ radians}

This gives us the golden conversion:

2π radians=360°\boxed{2\pi \text{ radians} = 360°}

Dividing both sides by 2:

π radians=180°\boxed{\pi \text{ radians} = 180°}

This single fact is the key to all conversions.

Converting Degrees → Radians

To convert degrees to radians, multiply by π180\dfrac{\pi}{180}:

Radians=Degrees×π180\text{Radians} = \text{Degrees} \times \frac{\pi}{180}

Examples:

DegreesWorkingRadians
90°90×π18090 \times \frac{\pi}{180}π2\frac{\pi}{2}
60°60×π18060 \times \frac{\pi}{180}π3\frac{\pi}{3}
45°45×π18045 \times \frac{\pi}{180}π4\frac{\pi}{4}
30°30×π18030 \times \frac{\pi}{180}π6\frac{\pi}{6}
270°270×π180270 \times \frac{\pi}{180}3π2\frac{3\pi}{2}

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