Form 5 Additional Mathematics Bab 1: Circular Measure
In circular measure, angles are measured in radians (rad) alongside degrees ($^\circ$).
One radian ($1\text{ rad}$) is defined as the measure of the angle subtended at the centre of a circle by an arc whose length is equal to the radius ($r$) of the circle.
Since a complete revolution around the centre of a circle subtends an angle of $360^\circ$ and corresponds to an arc length of $2\pi r$:
$$2\pi\text{ rad} = 360^\circ \quad \implies \quad \pi\text{ rad} = 180^\circ$$Note: Unless stated otherwise, take $\pi = 3.142$ or use the calculator's exact $\pi$ value.
When an angle $\theta$ at the centre of a circle of radius $r$ is measured in radians, the arc length $s$ is directly proportional to $\theta$:
$$s = r\theta$$where:
A segment is bounded by an arc and a chord. To find the perimeter of a minor segment:
$$\text{Perimeter of segment} = \text{Arc length } (s) + \text{Chord length } (c)$$Using trigonometry on the isosceles triangle formed by two radii and the chord:
$$\text{Chord length } (c) = 2r \sin\left(\frac{\theta}{2}\right)$$ $$\text{Perimeter of segment} = r\theta + 2r \sin\left(\frac{\theta}{2}\right)$$When the central angle $\theta$ is measured in radians, the area $A$ of a sector of a circle with radius $r$ is given by:
$$A = \frac{1}{2}r^2\theta$$The area of a segment bounded by an arc subtending angle $\theta$ (in radians) and a chord is calculated by subtracting the area of the isosceles triangle from the area of the sector:
$$\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}$$ $$\text{Area of segment} = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta = \frac{1}{2}r^2(\theta - \sin\theta)$$Real-world applications often involve combined regions, belts and pulleys, or geometric designs: