📚 Additional Mathematics (0606) Study Notes: Chapter 9 – Circular Measure

Welcome to Circular Measure! Don't worry, this chapter isn't as circular as it sounds. You are already an expert at measuring angles in degrees (like \(90^\circ\) or \(360^\circ\)), but in Additional Mathematics, we introduce a new, much more useful unit: the Radian.

Why do we need radians? Because when we use radians, the formulas for arc length and sector area become incredibly simple and elegant. These simple formulas are essential when you move on to Calculus and higher mathematics!

1. Understanding the Radian

A radian (often abbreviated as 'rad' or simply no unit given) is just another way to measure an angle, but unlike degrees, it is defined geometrically based on the radius of the circle.

Key Definition: The Radian

One radian is the angle subtended at the centre of a circle when the arc length is equal to the radius of the circle.

  • Imagine taking the radius (\(r\)) and bending it around the circumference. The angle it forms at the centre is exactly 1 radian.
Relating Radians and Degrees

If you wrap the radius around the circumference repeatedly, it takes exactly \(2\pi\) lengths of the radius to complete a full circle. This gives us our most important conversion factor:

A full circle is \(360^\circ\) or \(2\pi\) radians.

Therefore, the simpler, fundamental relationship is:

\[ \mathbf{180^\circ = \pi \text{ radians}} \]

(Remember: \(\pi\) is approximately 3.14159...)

Quick Review: Key Radian Facts
  • \(360^\circ = 2\pi\) rad
  • \(180^\circ = \pi\) rad (The straight line angle)
  • \(90^\circ = \frac{\pi}{2}\) rad (The right angle)

Key Takeaway: Radians link the angle directly to the radius and arc length, making calculations simpler.

2. Conversion Between Degrees and Radians

You must be fluent in converting between these two systems, especially when the exam question requires an answer in a specific unit.

2.1 Degrees to Radians

To convert from Degrees to Radians, you need to multiply by the factor \(\frac{\pi}{180}\).

\[ \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \]

Example: Convert \(60^\circ\) to radians.

\[ 60 \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians} \]

2.2 Radians to Degrees

To convert from Radians to Degrees, you need to multiply by the factor \(\frac{180}{\pi}\).

\[ \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi} \]

Example: Convert \(\frac{3\pi}{4}\) radians to degrees.

\[ \frac{3\pi}{4} \times \frac{180}{\pi} = \frac{3 \times 180}{4} = 3 \times 45^\circ = 135^\circ \]

💡 Memory Aid: Conversion Trick

If you want the answer to contain \(\pi\), put \(\pi\) on top (e.g., Degrees to Radians).
If you want the \(\pi\) to disappear, put \(\pi\) on the bottom (e.g., Radians to Degrees).

Common Mistake to Avoid: Never mix units in a single calculation! If you use the radian formulas in the next sections, your angle \(\theta\) must be in radians.

3. Calculating Arc Length (\(s\))

The arc length is the distance along the curved edge of the sector.

The Formula (Radian Measure)

When the angle \(\theta\) is measured in radians, the arc length \(s\) is given by:

\[ s = r\theta \]

Where:

  • \(s\) is the Arc Length
  • \(r\) is the Radius
  • \(\theta\) is the angle in Radians

Did you know? This formula is essentially the definition of a radian! If \(\theta = 1\) rad, then \(s = r\).

Step-by-Step Example: Arc Length

Problem: Find the arc length of a sector with radius \(6\) cm and a central angle of \(75^\circ\).

Step 1: Convert the angle to Radians.

\[ \theta = 75^\circ \times \frac{\pi}{180} = \frac{5\pi}{12} \text{ rad} \]

Step 2: Apply the Arc Length formula.

\[ s = r\theta = 6 \times \frac{5\pi}{12} = \frac{30\pi}{12} = \frac{5\pi}{2} \text{ cm} \]

Key Takeaway: The arc length formula is simply \(s = r\theta\). Memorize this, as it is not given on the exam formula sheet.

4. Calculating Area of a Sector (\(A\))

A sector is the pizza-slice shape cut out of a circle.

The Formula (Radian Measure)

When the angle \(\theta\) is measured in radians, the area \(A\) of the sector is given by:

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

Where:

  • \(A\) is the Area of the Sector
  • \(r\) is the Radius
  • \(\theta\) is the angle in Radians
⚠️ IMPORTANT MEMORY CHECK

Both primary formulas must be memorized:

  • Arc Length: \(s = r\theta\)
  • Sector Area: \(A = \frac{1}{2} r^2 \theta\)

Notice the structure: Arc length is proportional to \(r\) (like perimeter), while Area is proportional to \(r^2\).

Step-by-Step Example: Sector Area

Problem: A sector has an angle of \(0.8\) radians and a radius of \(5\) m. Find the area.

Step 1: Check units. The angle is already in radians (\(0.8\)).

Step 2: Apply the Area formula.

\[ A = \frac{1}{2} r^2 \theta = \frac{1}{2} (5)^2 (0.8) \]

\[ A = \frac{1}{2} (25) (0.8) = 12.5 \times 0.8 = 10 \text{ m}^2 \]

Key Takeaway: Sector area uses the simple formula \(A = \frac{1}{2} r^2 \theta\), provided \(\theta\) is in radians.

5. Solving Problems with Compound Shapes

Many exam questions involve combining sectors with other shapes, most commonly triangles, to find the area of a Segment or the perimeter of a complex figure.

5.1 The Area of a Segment

A segment is the area enclosed by an arc and the chord connecting the endpoints of that arc. To find the segment area, you must subtract the triangle area from the sector area.

Area of Segment = Area of Sector - Area of Triangle

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

2. Area of Triangle: We use the formula for a non-right-angled triangle: \(A = \frac{1}{2} ab \sin C\). Since \(a\) and \(b\) are both radii (\(r\)), and \(C\) is the central angle \(\theta\):

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

3. Area of Segment:

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

Note: For the term \(\frac{1}{2} r^2 \theta\), \(\theta\) MUST be in radians. For the term \(\frac{1}{2} r^2 \sin \theta\), \(\theta\) can be in degrees or radians, but you must ensure your calculator is set to the correct mode for calculating \(\sin \theta\)! It is usually safest to use radians throughout the problem if the question uses radians.

5.2 Perimeter of Compound Shapes

To find the perimeter, you simply add up all the side lengths bounding the shape.

  • The curved part will be the Arc Length (\(s = r\theta\)).
  • The straight edges might be radii (\(r\)), chords, or external lines.

If you need to find the length of the chord (\(c\)) that subtends the angle \(\theta\), you can use the cosine rule on the triangle formed by the two radii and the chord:

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

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

Remember: Use your basic geometry knowledge (isosceles triangles, right angles, trigonometry) alongside the new circular measure formulas.

Key Takeaway: Compound shape problems usually involve subtracting the area of a triangle (\(\frac{1}{2} r^2 \sin \theta\)) from the area of a sector (\(\frac{1}{2} r^2 \theta\)).

Quick Review Checklist for Circular Measure

📝 Essential Knowledge Check
  • Conversion: \(180^\circ = \pi\) radians.
  • Arc Length Formula (MUST know): \(s = r\theta\) (\(\theta\) in radians).
  • Sector Area Formula (MUST know): \(A = \frac{1}{2} r^2 \theta\) (\(\theta\) in radians).
  • Triangle Area within Sector: \(A_{\text{tri}} = \frac{1}{2} r^2 \sin \theta\).
  • Segment Area: \(\frac{1}{2} r^2 \theta - \frac{1}{2} r^2 \sin \theta\).
  • Units: Always check if your answer needs to be left in terms of \(\pi\) (exact value) or as a decimal (to 3 s.f.).

You've got this! Circular measure is straightforward once you master the radian and keep your formulas straight.