When the angle at the centre of a circle is measured in radians, the arc length can be found simply using:
\[ \text{Arc length } (l) = r\theta \] where:
â This formula comes directly from the definition of radians. For a full revolution, \( \theta = 2\pi \), so \( l = 2\pi r \) (the circumference).
The area of a sector of a circle with angle \( \theta \) (in radians) is:
\[ \text{Sector area } (A) = \frac{1}{2} r^2 \theta \] since the fraction of the circle covered by the angle \( \theta \) is \( \theta / (2\pi) \).
A segment of a circle is the region between a chord and the arc it subtends. To find its area:
\[ \text{Segment area} = \text{Sector area} - \text{Triangle area} \] \[ = \frac{1}{2} r^2 \theta - \frac{1}{2} r^2 \sin\theta \] or equivalently \[ = \frac{1}{2} r^2 (\theta - \sin\theta), \] where \( \theta \) is the angle at the centre in radians.
\[ l = r\theta = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} \ \text{cm}. \]
\[ A = \frac{1}{2} r^2 \theta = \frac{1}{2} \times 10^2 \times 2 = 100 \ \text{cm}^2. \]
\[ \text{Segment area} = \frac{1}{2} r^2 (\theta - \sin\theta) = \frac{1}{2} \times 6^2 \times (1.2 - \sin 1.2) \] \[ = 18 \times (1.2 - 0.9320) = 18 \times 0.268 \approx 4.82 \ \text{cm}^2. \]