Welcome to Radian Measure!

In your mathematical journey so far, you have likely used degrees to measure angles. We know that a right angle is \(90^\circ\) and a full circle is \(360^\circ\). But as you move into AS Level Mathematics, specifically Unit PSM1, you will discover a more "natural" way to measure angles: the Radian.

Radians make formulas for circles much simpler and are essential for higher-level calculus. Don't worry if it feels strange at first—by the end of these notes, you'll see why mathematicians love them!

1. What is a Radian?

Imagine you have a circle with a radius \(r\). If you take a piece of string the exact same length as the radius and wrap it around the edge of the circle (the arc), the angle formed at the center is exactly 1 radian.

Because the circumference of a circle is \(2\pi r\), there are exactly \(2\pi\) radians in a full circle.

The Big Rule:
\(180^\circ = \pi \text{ radians}\)
\(360^\circ = 2\pi \text{ radians}\)

How to Convert Between Degrees and Radians

If you ever get stuck, just remember that \(\pi\) and \(180^\circ\) are the same thing!

  • Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
  • Radians to Degrees: Multiply by \(\frac{180}{\pi}\)

Quick Review: Common Angles

\(30^\circ = \frac{\pi}{6}\)
\(45^\circ = \frac{\pi}{4}\)
\(60^\circ = \frac{\pi}{3}\)
\(90^\circ = \frac{\pi}{2}\)
\(180^\circ = \pi\)

2. Arc Length

An arc is just a portion of the circumference (the "crust" of a pizza slice). When the angle \(\theta\) is measured in radians, the formula for the length of the arc \(l\) is incredibly simple.

The Formula:
\(l = r\theta\)

Where:
\(l\) is the arc length
\(r\) is the radius
\(\theta\) is the angle in radians

Example:

Find the arc length of a circle with radius \(5\text{ cm}\) and a central angle of \(\frac{\pi}{4}\) radians.
\(l = 5 \times \frac{\pi}{4} = \frac{5\pi}{4} \approx 3.93\text{ cm}\)

Key Takeaway: Never use this formula with degrees! If the question gives you \(60^\circ\), you must convert it to \(\frac{\pi}{3}\) radians first.

3. Area of a Sector

A sector is a "slice" of the circle (the whole pizza slice). Again, using radians makes our formula much cleaner than the one you used in GCSE.

The Formula:
\(A = \frac{1}{2}r^2\theta\)

Where:
\(A\) is the area of the sector
\(r\) is the radius
\(\theta\) is the angle in radians

Example:

A sector has a radius of \(8\text{ cm}\) and an angle of \(1.2\) radians. Find its area.
\(A = \frac{1}{2} \times 8^2 \times 1.2\)
\(A = 0.5 \times 64 \times 1.2 = 38.4\text{ cm}^2\)

Did you know? This formula is just a fraction of the total area. Since a full circle is \(2\pi\) radians, the area of a sector is \(\frac{\theta}{2\pi} \times \pi r^2\). The \(\pi\) symbols cancel out, leaving you with \(\frac{1}{2}r^2\theta\)!

4. Common Exam Pitfalls to Avoid

The "Perimeter" Trap

Exam questions often ask for the perimeter of a sector. Students often calculate the arc length (\(l\)) and stop there.
Remember: The perimeter includes the arc plus the two straight sides (the radii).
Perimeter of a sector = \(r\theta + 2r\)

Calculator Mode

This is the most common mistake in Unit PSM1!
If you are calculating \(\sin(\theta)\) or \(\cos(\theta)\) where \(\theta\) is in radians, your calculator must be in RAD mode. If you are working with degrees, it must be in DEG mode. Always check the top of your screen for a little 'R' or 'D'.

Working with \(\pi\)

Unless the question asks for a decimal answer (usually to 3 significant figures), it is often better to keep \(\pi\) in your working. This keeps your answer exact. For example, \(\frac{5\pi}{3}\) is more accurate than \(5.24\).

5. Summary Checklist

  • Can I convert between degrees and radians using the factor \(\frac{\pi}{180}\)?
  • Do I remember that \(l = r\theta\) only works if \(\theta\) is in radians?
  • Do I remember that \(A = \frac{1}{2}r^2\theta\) only works if \(\theta\) is in radians?
  • Have I checked if the question asks for Area or Perimeter?
  • Is my calculator in the correct mode for the units I am using?

Note: For further applications of these formulas in triangles (like the Sine or Cosine rule), see the next chapter: "Trigonometry: rules, graphs and equations".