Arc Length and Sector Area for the TMUA

Introduction to Radian Measure

While degrees are a common method for measuring angles, where one full revolution is $360^{\circ}$, the number 360 is somewhat arbitrary. Radians provide a more natural measure for angles, particularly for use in higher mathematics and calculus. A radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

In a circle of radius 1, an angle of 1 radian corresponds to an arc length of exactly 1. Consequently, because the circumference of a circle with radius 1 is $2\pi$, a complete revolution is equal to $2\pi$ radians. Therefore, 1 radian is approximately equal to $\frac{360}{2\pi} \approx 57.3^{\circ}$.

Converting Between Degrees and Radians

To convert between these two systems, we use the fact that $360^{\circ} = 2\pi$ radians.

1. To convert $\theta$ degrees to radians, calculate the fraction of a full revolution and multiply by $2\pi$: $\theta \text{ degrees} = \frac{\theta}{360} \times 2\pi \text{ radians}$.
2. To convert $\alpha$ radians to degrees, calculate the fraction of a full revolution and multiply by 360: $\alpha \text{ radians} = \frac{\alpha}{2\pi} \times 360 \text{ degrees}$.

You should memorise the following standard conversions:

• $30^{\circ} = \frac{\pi}{6}$ radians.
• $45^{\circ} = \frac{\pi}{4}$ radians.
• $60^{\circ} = \frac{\pi}{3}$ radians.
• $90^{\circ} = \frac{\pi}{2}$ radians.
• $180^{\circ} = \pi$ radians.
• $360^{\circ} = 2\pi$ radians.

Arc Length and Sector Area Formulae

When using radians, the formulae for the geometry of a circle become remarkably simple. For a sector of a circle with radius $r$ and angle $\alpha$ radians:

• Arc length $= r\alpha$
• Area of sector $= \frac{1}{2}r^2\alpha$

Proof of the Arc Length Formula

We know that an angle of $\alpha$ radians represents a specific fraction of a whole circle. Since a whole circle is $2\pi$ radians, the fraction is $\frac{\alpha}{2\pi}$. The arc length is therefore:

$\text{Arc length} = \text{Circumference} \times \text{fraction of circle}$

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

Proof of the Sector Area Formula

Similarly, the area of a sector is the area of the entire circle multiplied by the fraction of the circle that the sector occupies:

$\text{Area of sector} = \text{Area of whole circle} \times \text{fraction of circle}$

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

These proofs demonstrate why radians are the preferred measure: they eliminate the arbitrary constants associated with degrees and link linear and angular measures directly through the radius.