Hello Future A-Level Mathematician!

Welcome to the chapter on Circular Measure! Don't worry if this sounds a bit abstract. This topic is simply about a new, incredibly useful way to measure angles, replacing the degrees you learned in IGCSE. It's essential because it links geometry (shapes) directly with calculus (rates of change), which is fundamental to higher mathematics.

By the end of these notes, you’ll be able to confidently handle angles in radians and calculate the sizes of curved parts of circles, like the crust of a pizza slice!


1. Understanding the Radian: A Natural Unit

1.1 What is a Radian?

In school, you learned that a full turn is \(360^\circ\). But why 360? It’s a historical choice (it divides nicely). Mathematicians prefer a unit based on the geometry of the circle itself: the Radian.

Definition of One Radian (1 rad)

One radian is the angle subtended at the centre of a circle when the arc length \(s\) is exactly equal to the radius \(r\) of the circle.

Imagine taking a piece of string exactly the length of the radius and laying it along the curve of the circle's edge. The angle you make by drawing lines from the centre to the ends of that string is 1 radian.

The Key Relationship: Radians vs. Degrees

The circumference of a circle is \(C = 2\pi r\). Since the angle for a full turn is measured by how many radii fit around the circle, we have the core relationship:

\(360^\circ = 2\pi\) radians

This means that:

\(180^\circ = \pi\) radians

Converting Between Radians and Degrees

We use the relationship \(180^\circ = \pi\) radians to create simple conversion factors:

  • To convert Degrees to Radians: Multiply the degrees by \(\frac{\pi}{180}\).
    Example: \(90^\circ = 90 \times \frac{\pi}{180} = \frac{\pi}{2}\) rad.
  • To convert Radians to Degrees: Multiply the radians by \(\frac{180}{\pi}\).
    Example: \(\frac{\pi}{3}\) rad \( = \frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ\).

! CRITICAL WARNING !

When solving trigonometry problems involving circular measure, ensure your calculator is set to Radian Mode. Using degrees in a radian problem is the most common mistake students make in this chapter!

Quick Review: Key Angles in Radians

Memorise these common conversions; they save time in the exam:

  • \(360^\circ = 2\pi\)
  • \(180^\circ = \pi\)
  • \(90^\circ = \frac{\pi}{2}\)
  • \(60^\circ = \frac{\pi}{3}\)
  • \(45^\circ = \frac{\pi}{4}\)
  • \(30^\circ = \frac{\pi}{6}\)

***

Key Takeaway (Radian): Radians are the "natural" measurement for angles in A-Level Maths, based on the radius. Always remember \(\pi\) radians is \(180^\circ\).


2. Arc Length of a Sector

2.1 The Arc Length Formula

The Arc Length (\(s\)) is the distance along the curved edge of a circular sector.

The formula for arc length is beautifully simple, provided the angle \(\theta\) is measured in radians:

\[s = r\theta\]

Where:

\(s\) is the arc length.
\(r\) is the radius.
\(\theta\) is the angle subtended at the centre (must be in radians).

Step-by-Step Example: Finding Arc Length

Problem: A circle has a radius of 5 cm. Find the arc length subtended by an angle of \(1.2\) radians.

  • Step 1: Check Units. The angle \(\theta = 1.2\) is already in radians.
  • Step 2: Identify \(r\). \(r = 5\) cm.
  • Step 3: Apply Formula. \(s = r\theta\)
    \(s = 5 \times 1.2\)
  • Step 4: Calculate. \(s = 6\) cm.
What if the angle is given in degrees?

You must convert it first!

Example: Find the arc length if \(r = 10\) cm and \(\theta = 72^\circ\).

  • Step 1: Convert to Radians. \(\theta = 72 \times \frac{\pi}{180} = \frac{2\pi}{5}\) rad.
  • Step 2: Apply Formula. \(s = 10 \times \frac{2\pi}{5}\)
  • Step 3: Calculate. \(s = 4\pi \approx 12.6\) cm (3 s.f.).

***

Key Takeaway (Arc Length): \(s = r\theta\). Always ensure \(\theta\) is in radians before multiplying!


3. Area of a Sector

3.1 The Sector Area Formula

The Area of a Sector (\(A\)) is the space enclosed by two radii and the arc (the slice of the circle).

The formula mirrors the arc length formula, but involves \(r^2\):

\[A = \frac{1}{2}r^2\theta\]

Where:

\(A\) is the sector area.
\(r\) is the radius.
\(\theta\) is the angle subtended at the centre (must be in radians).

Did you know? If you use the full circle angle (\(2\pi\)) in this formula, you get \(A = \frac{1}{2}r^2(2\pi) = \pi r^2\), which is the standard area of a circle! This shows why radians are so natural.

Step-by-Step Example: Finding Sector Area

Problem: A sector has a radius of 8 metres and a central angle of \(\frac{\pi}{4}\) radians. Find its area.

  • Step 1: Check Units. \(\theta = \frac{\pi}{4}\) rad.
  • Step 2: Identify \(r\). \(r = 8\) m.
  • Step 3: Apply Formula. \(A = \frac{1}{2}r^2\theta\)
    \(A = \frac{1}{2}(8^2) \left(\frac{\pi}{4}\right)\)
  • Step 4: Calculate. \(A = \frac{1}{2}(64) \left(\frac{\pi}{4}\right) = 32 \times \frac{\pi}{4} = 8\pi\) m\(^2\).
    (If required to 3 s.f.: \(8\pi \approx 25.1\) m\(^2\).)

***

Key Takeaway (Sector Area): \(A = \frac{1}{2}r^2\theta\). Again, radians are non-negotiable for this formula.


4. Solving Complex Problems: Segments and Perimeter

In examination questions, you often need to combine the circular measure formulas with your knowledge of basic geometry, especially triangles.

4.1 Calculating the Area of a Segment

A segment is the area enclosed by an arc and the chord connecting the arc's endpoints. (It's the "crust" area remaining after cutting off the triangular slice.)

Area of Segment = Area of Sector – Area of Triangle

Step-by-Step Formula for Segment Area

Consider a sector OAB with radius \(r\) and central angle \(\theta\) (in radians). The triangle inside is OAB.

1. Area of Sector OAB: \(A_{\text{sector}} = \frac{1}{2}r^2\theta\)

2. Area of Triangle OAB: Since the sides adjacent to the angle \(\theta\) are both radii \(r\), we use the standard trigonometric area formula: \(A_{\text{triangle}} = \frac{1}{2}ab \sin C\)

\[A_{\text{triangle}} = \frac{1}{2}r^2 \sin \theta\]

3. Area of Segment:

\[A_{\text{segment}} = \frac{1}{2}r^2\theta - \frac{1}{2}r^2 \sin \theta\]

Note: For the \( \frac{1}{2}r^2 \sin \theta \) part, your calculator can technically be in Degree or Radian mode, but since the rest of the calculation is in radians, it is safest to keep your calculator in Radian Mode for the entire calculation.

4.2 Calculating Perimeters of Composite Shapes

The perimeter of a shape involving an arc is simply the sum of all its boundary lengths.

Perimeter of Sector OAB = Arc length \(s\) + Radius \(r\) + Radius \(r\)

\[P_{\text{sector}} = r\theta + 2r\]

Perimeter of Segment = Arc length \(s\) + Chord length

To find the Chord Length (the straight line across the segment), you must use the Cosine Rule on the triangle OAB:

\[c^2 = r^2 + r^2 - 2(r)(r) \cos \theta\]

\[c = \sqrt{2r^2 (1 - \cos \theta)}\]

Don't worry about memorising the complex chord formula; just remember to use the Cosine Rule on the triangle OAB.

***

Key Takeaway (Segments): Segment Area is Sector Area minus Triangle Area. Use standard P1 geometry (Cosine Rule or Area \(\frac{1}{2}ab \sin C\)) where needed, but keep angles consistent.

5. Quick Review and Study Checklist

5.1 Essential Formulae (from MF19)

Make sure these are second nature:

  • Angle Conversion: \(\pi\) radians = \(180^\circ\)
  • Arc Length: \(s = r\theta\) (\(\theta\) in radians)
  • Area of Sector: \(A = \frac{1}{2}r^2\theta\) (\(\theta\) in radians)
  • Area of Triangle (Non-Right Angle): \(A = \frac{1}{2}ab \sin C\)

5.2 Common Mistakes to Avoid

1. Forgetting to Check Calculator Mode

If you use \(s=r\theta\) or \(A=\frac{1}{2}r^2\theta\) while the calculator is in Degree Mode, your answer will be wildly wrong. Always verify Radian Mode is active.

2. Mixing Arc Length and Sector Area

Arc length \(s\) is a linear measure (power of \(r\) is 1). Area \(A\) is a squared measure (power of \(r\) is 2). If you confuse the formulae, the units will tell you:

\(s = r\theta\) (cm)
\(A = \frac{1}{2}r^2\theta\) (cm\([^2]\))

3. Using Degrees in the Formulae

If a question provides an angle in degrees, the first step must be conversion using \(\times \frac{\pi}{180}\) before using the standard circular measure formulae.


You've Got This!

Circular measure is a fundamental topic that paves the way for advanced calculus and trigonometry. Master the radian, trust your formulas, and always remember to check your calculator mode!