Radian Measure and Sector Geometry

Introduction to Radian Measure

Usually, the first method for measuring angles you encounter is using degrees, where one full revolution is $360$ degrees. There is nothing inherently special about the number $360$: some suggests it relates to the approximate number of days in a year, but other units exist, such as Gradians where a right angle is $100$ units. These measures are somewhat arbitrary. However, there is one measure for angles that is more natural than all others: radians.

We define one radian as the angle subtended by a sector of a circle of radius $1$ when the arc length is also exactly $1$.

Equivalently, because the circumference of a circle of radius $1$ is $2\pi$, a full revolution is equal to $2\pi$ radians. Therefore, $1$ radian is approximately equal to $\frac{360}{2\pi} \approx 57.298^\circ$. While the symbol for radians is a superscript c ($1^c$), it is common to simply write rad or use the number alone.

Converting Between Degrees and Radians

Conversion between degrees and radians is straightforward because $360$ degrees corresponds to $2\pi$ radians.

To convert $\theta$ degrees to radians:

Since $\theta$ degrees is the fraction $\frac{\theta}{360}$ of a full revolution, the radian measure is:

$\theta \text{ degrees} = \frac{\theta}{360} \times 2\pi \text{ radians}$

To convert $\alpha$ radians to degrees:

Similarly, $\alpha$ radians is the fraction $\frac{\alpha}{2\pi}$ of a full revolution, so:

$\alpha \text{ radians} = \frac{\alpha}{2\pi} \times 360 \text{ degrees}$

Students are expected to know these standard conversions by heart:

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

Arc Length and Sector Area

You must understand and be able to use the formulae for arc length and sector area when the angle $\alpha$ is measured in radians.

$arc\ length = r\alpha$

$area\ of\ sector = \frac{1}{2}r^2\alpha$

To prove these formulae, we recall that an angle $\alpha$ in radians represents the fraction $\frac{\alpha}{2\pi}$ of a full circle.

Proof of Arc Length:

The arc length is the circumference multiplied by the fraction of the circle corresponding to the arc:

$arc\ length = 2\pi r \times \frac{\alpha}{2\pi} = r\alpha$

Proof of Sector Area:

The area of the sector is the total area of the circle multiplied by the fraction of the circle that makes up the sector:

$area\ of\ sector = \pi r^2 \times \frac{\alpha}{2\pi} = \frac{1}{2}r^2\alpha$

Area of a Segment

Although the guide focuses on sectors, the TMUA specification requires the use of radian measure for segments. A segment is the region bounded by a chord and an arc. To find the area of a segment, you subtract the area of the triangle formed by the two radii and the chord from the area of the full sector.

Using the formula for the area of a triangle $\frac{1}{2}ab \sin C$ where the sides are both the radius $r$:

$Area\ of\ segment = Area\ of\ sector - Area\ of\ triangle$

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