Circular Measure and Radian

Subject: Additional Mathematics
Topic: 6
Cambridge Code: 4037 / 0606

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Radian Measure

Radian - Unit of angle measurement based on arc length

Definition

One radian is the angle subtended at the center of a circle when the arc length equals the radius

$1 \text{ radian} = \frac{\text{arc length}}{\text{radius}}$

Converting Between Degrees and Radians

$\pi \text{ radians} = 180°$

$1 \text{ radian} = \frac{180°}{\pi} \approx 57.3°$

$1° = \frac{\pi}{180} \text{ radians}$

Conversion Formulas: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

Common Angles

Degrees	Radians
0°	0
30°	$\frac{\pi}{6}$
45°	$\frac{\pi}{4}$
60°	$\frac{\pi}{3}$
90°	$\frac{\pi}{2}$
180°	$\pi$
360°	$2\pi$

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Arc Length

Arc Length - Distance along the circumference

Formula (with angle in radians)

$s = r\theta$

where:

• $s$ is arc length
• $r$ is radius
• $\theta$ is angle in radians

Example

Find arc length when radius is 5 cm and angle is $\frac{\pi}{3}$ radians:

$s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} \approx 5.24 \text{ cm}$

Formula (with angle in degrees)

$s = \frac{\theta}{360°} \times 2\pi r = \frac{\theta\pi}{180} \times r$

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Sector Area

Sector - Region bounded by two radii and an arc

Formula (with angle in radians)

$A = \frac{1}{2}r^2\theta$

where:

• $A$ is sector area
• $r$ is radius
• $\theta$ is angle in radians

Example

Find sector area when radius is 4 m and angle is $\frac{\pi}{6}$ radians:

$A = \frac{1}{2}(4)^2 \times \frac{\pi}{6} = \frac{1}{2} \times 16 \times \frac{\pi}{6} = \frac{8\pi}{3} \approx 8.38 \text{ m}^2$

Formula (with angle in degrees)

$A = \frac{\theta}{360°} \times \pi r^2$

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Segment Area

Segment - Region bounded by a chord and an arc

Formula

$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$

$= \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta$

$= \frac{1}{2}r^2(\theta - \sin\theta)$

where $\theta$ is in radians

Example

Find segment area when $r = 6$ cm and $\theta = \frac{\pi}{3}$:

$A = \frac{1}{2}(36)\left(\frac{\pi}{3} - \sin\frac{\pi}{3}\right)$

$= 18\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right)$

$= 6\pi - 9\sqrt{3} \approx 2.18 \text{ cm}^2$

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Angles in Standard Position

Standard Position - Angle measured counterclockwise from positive x-axis

Coterminal Angles

Angles that differ by multiples of $2\pi$ radians (or 360°)

Example: $\frac{\pi}{4}$ and $\frac{\pi}{4} + 2\pi = \frac{9\pi}{4}$ are coterminal

Reference Angle

Acute angle between terminal side and x-axis

Used to find trigonometric ratios in any quadrant

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Circular Functions

Angular Velocity - Rate of change of angle

$\omega = \frac{\theta}{t} \text{ (radians per unit time)}$

Linear Velocity: $v = r\omega$

Example

A wheel of radius 30 cm rotates at 20 rpm (revolutions per minute).

Find angular velocity in rad/s: $\omega = 20 \text{ rev/min} = 20 \times \frac{2\pi}{60} = \frac{2\pi}{3} \text{ rad/s} \approx 2.09 \text{ rad/s}$

Find linear velocity of point on rim: $v = r\omega = 0.3 \times \frac{2\pi}{3} = 0.2\pi \text{ m/s} \approx 0.628 \text{ m/s}$

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Key Points to Remember

1. Radian is more natural for circles than degrees
2. $\pi$ radians = 180°
3. Arc length: $s = r\theta$ (angle in radians)
4. Sector area: $A = \frac{1}{2}r^2\theta$ (angle in radians)
5. Segment area: $A = \frac{1}{2}r^2(\theta - \sin\theta)$
6. Angles in standard position measured counterclockwise

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Worked Examples

Example 1: Convert Degrees to Radians

Convert 225° to radians: $225° = 225° \times \frac{\pi}{180°} = \frac{225\pi}{180} = \frac{5\pi}{4} \text{ radians}$

Example 2: Arc Length

A circle has radius 8 cm. Find the arc length of a sector with angle $\frac{2\pi}{5}$ radians:

$s = r\theta = 8 \times \frac{2\pi}{5} = \frac{16\pi}{5} \approx 10.05 \text{ cm}$

Example 3: Sector Area

Find the area of a sector with radius 6 m and angle 120°:

First convert: $120° = \frac{2\pi}{3}$ radians

$A = \frac{1}{2}r^2\theta = \frac{1}{2}(36)\frac{2\pi}{3} = 12\pi \approx 37.7 \text{ m}^2$

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Practice Questions

1. Convert to radians:

   • 60°
   • 150°
   • 315°

2. Convert to degrees:

   • $\frac{\pi}{2}$
   • $\frac{3\pi}{4}$
   • $\frac{5\pi}{6}$

3. Find arc length and sector area for:

   • $r = 5$ cm, $\theta = \frac{\pi}{6}$
   • $r = 10$ m, $\theta = 1.5$ radians

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Revision Tips

• Practice converting between degrees and radians
• Remember: $s = r\theta$ uses angle in radians
• Sector area: $A = \frac{1}{2}r^2\theta$ (radians)
• Segment area formula includes subtraction of triangle
• Draw diagrams to visualize sectors and segments