¶Angles and Circular Measurement

Angles measure the amount of rotation between two intersecting lines. While everyday applications use degrees, advanced mathematics and physics rely heavily on radians.

¶Radians

A radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the circle's radius. Because the total circumference of a circle is $C = 2\pi r$, there are exactly $2\pi$ radians in a full $360^\circ$ rotation.

To convert between them: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180^\circ}$ $\text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi}$

¶Arc Length

The arc length ($s$) is the distance along the curved edge of a circle. When the angle $\theta$ is measured in radians, the formula is elegantly simple: $s = r \cdot \theta$

Why? If $\theta = 2\pi$ (a full circle), the formula gives $s = r(2\pi) = 2\pi r$, which is the circumference.

¶Sector Area

A sector is a portion of a circle enclosed by two radii and an arc (like a slice of pizza). The area of a sector is proportional to its central angle. $A = \frac{1}{2} r^2 \theta$

Why? If $\theta = 2\pi$, the formula gives $A = \frac{1}{2} r^2 (2\pi) = \pi r^2$, the area of the full circle.

¶References

• Khan Academy: Radians & Arc Length
• Stewart Calculus (Early Transcendentals)