SPM Paper 1 & 2 Core Topic

Radian Measure, Arc Length, Segment Area & Pulley Systems

Circular measure introduces radians as the natural mathematical angle measurement. Master arc length ($s = r\theta$), sector area ($A = \frac{1}{2}r^2\theta$), segment area formulas, perimeters of composite geometric figures, and industrial pulley-belt conveyor modeling.

1. Core Circular Measure Formulas

Radian Conversion

$\theta\text{ rad} = \theta^\circ \times \frac{\pi}{180^\circ}$
$\theta^\circ = \theta\text{ rad} \times \frac{180^\circ}{\pi}$

Unless stated otherwise, use $\pi = 3.142$ or calculator $\pi$.

Arc Length ($s$)

$s = r\theta$

Chord length: $c = 2r\sin\left(\frac{\theta}{2}\right)$ (or Cosine Rule $c^2 = 2r^2(1 - \cos\theta)$).

Sector & Segment Area

$A_{\text{sector}} = \frac{1}{2}r^2\theta$
$A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$

$\theta$ MUST be in radians!

2. SPM Examiner Pitfalls in Circular Measure

Trap 1: Evaluating $\sin\theta$ in Degree Mode When $\theta$ is in Radians:

In the segment formula $A = \frac{1}{2}r^2(\theta - \sin\theta)$, the first $\theta$ is purely in radians, while $\sin\theta$ requires your calculator to be either set to RAD mode or converted to degrees first ($\theta \times \frac{180^\circ}{\pi}$). Computing $\sin(1.2)$ in DEG mode produces $\sin(1.2^\circ) = 0.0209$ instead of $\sin(1.2\text{ rad}) = 0.9320$!

Trap 2: Major vs Minor Sector Confusion:

Pay close attention to whether the problem references the minor arc ($0 < \theta < \pi$) or the major arc ($\pi < \theta < 2\pi$). Reflex angle $\theta_{\text{major}} = 2\pi - \theta_{\text{minor}}$.

3. Progressive Worked Examples with SPM Marking Rubrics

Example 1 • Segment Area & Perimeter [5 Marks]
Diagram below shows a sector $AOB$ of a circle with center $O$ and radius $10\text{ cm}$. The subtended angle $\angle AOB$ is $1.2\text{ radians}$.

(a) Calculate the perimeter, in $\text{cm}$, of the shaded segment bounded by chord $AB$ and arc $AB$. [3 marks]

(b) Calculate the area, in $\text{cm}^2$, of the shaded segment. [2 marks]

Arc length $s = r\theta = 10(1.2) = 12.0\text{ cm}$. [1m: K1: Compute arc length]
Chord length $AB = 2r\sin\left(\frac{\theta}{2}\right) = 2(10)\sin(0.6\text{ rad}) = 20(0.56464) = 11.29\text{ cm}$. [1m: K1: Compute chord length]
$\text{Perimeter of segment} = s + AB = 12.0 + 11.29 = 23.29\text{ cm}$. [1m: N1: Correct perimeter]
(b) $\text{Area of segment} = \frac{1}{2}r^2(\theta - \sin\theta) = \frac{1}{2}(10^2)(1.2 - \sin(1.2\text{ rad}))$. [1m: K1: Segment formula]
$= 50(1.2 - 0.93204) = 50(0.26796) = 13.40\text{ cm}^2$. [1m: N1: Correct area]
Example 2 • KBAT Mechanical Pulley Conveyor Belt [6 Marks]
Two circular pulleys with centers $O_1$ and $O_2$ have radii $12\text{ cm}$ and $4\text{ cm}$ respectively. The distance between their centers is $O_1O_2 = 20\text{ cm}$. A tight conveyor belt connects the two pulleys externally touching at tangents $A$ and $B$.

(a) Show that the angle $\theta$ subtended at the center of the larger pulley by the tangent line is $\theta = 1.159\text{ radians}$. [2 marks]

(b) Calculate the straight length of the belt between contact points $AB$. [2 marks]

(c) Calculate the total length of the conveyor belt. [2 marks]

(a) Draw line from $O_2$ perpendicular to radius $O_1A$ at point $N$: $O_1N = r_1 - r_2 = 12 - 4 = 8\text{ cm}$. Hypotenuse $O_1O_2 = 20\text{ cm}$. [1m: K1: Right-angled triangle construction]
$\cos \alpha = \frac{8}{20} = 0.4 \implies \alpha = \cos^{-1}(0.4) = 1.1593\text{ rad} \approx 1.159\text{ radians}$. (Shown) [1m: N1]
(b) Straight length $AB = NO_2 = \sqrt{20^2 - 8^2} = \sqrt{400 - 64} = \sqrt{336} = 18.33\text{ cm}$. [2m: K1, N1: $AB = 18.33\text{ cm}$]
(c) Major arc on large pulley: angle $= 2\pi - 2(1.159) = 6.283 - 2.318 = 3.965\text{ rad}$. Arc length $= 12(3.965) = 47.58\text{ cm}$. Minor arc on small pulley: angle $= 2(1.159) = 2.318\text{ rad}$. Arc length $= 4(2.318) = 9.27\text{ cm}$. [1m: K1: Arc lengths of both pulleys]
Total Belt Length $= 2(AB) + \text{Arc}_1 + \text{Arc}_2 = 2(18.33) + 47.58 + 9.27 = 36.66 + 56.85 = 93.51\text{ cm}$. [1m: N1: Total belt length]
Open Form 5 Chapter 1 Worksheet