🎯 Circles in Action: Mastering Arc Length & Sector Areas!

**Why does this matter?** Every time you measure a slice of pizza or the curved edge of a racetrack, you are meeting *arc length*, *sector area*, and *segment area*. Mastering these circle measures not only boosts your O-Level score but also builds intuition for real-world design, engineering, and science problems!

Key Formulas & Concepts

Arc Length ( $L$ ) of angle $\theta$ (in radians) in a circle of radius $r$ :
$L = r\theta$
Sector Area ( $A_{\text{sector}}$ ):
$A_{\text{sector}} = \frac{1}{2}r^{2}\theta$
Segment Area ( $A_{\text{segment}}$ ):
$A_{\text{segment}} = A_{\text{sector}} - \frac{1}{2}r^{2}\sin\theta$
(subtract triangle area)
Convert degrees to radians:
$\theta_{\text{rad}} = \theta_{\degree} \times \frac{\pi}{180}$

Worked Example: Pizza Slice Geometry

Problem: A circular pizza has radius

14\text{ cm}

. A slice makes a central angle of

45^{\circ}

. Find (i) the arc length, (ii) the sector area, and (iii) the segment area (curved slice minus triangle).

Solution:
1. **Convert angle**:

45^{\circ} = 45 \times \frac{\pi}{180} = \frac{\pi}{4}

rad.
2. **Arc length**:

L = r\theta = 14 \times \frac{\pi}{4} = 3.5\pi \approx 11.0\text{ cm}

3. **Sector area**:

A_{\text{sector}} = \frac{1}{2}r^{2}\theta = \frac{1}{2}\times 14^{2} \times \frac{\pi}{4} = 49\pi \approx 154\text{ cm}^2

4. **Triangle area** inside the sector:

A_{\triangle} = \frac{1}{2}r^{2}\sin\theta = \frac{1}{2}\times 14^{2}\times \sin\frac{\pi}{4} = 98\times\frac{\sqrt{2}}{2} = 49\sqrt{2}\text{ cm}^2

5. **Segment area**:

A_{\text{segment}} = A_{\text{sector}} - A_{\triangle} = 49\pi - 49\sqrt{2} \approx 154 - 69.3 \approx 84.7\text{ cm}^2

**Keep practising!** Draw different sectors, change angles, and verify your calculations. Mastery comes from repetition—and your exam success will follow.