📏 Arc Length, Sector Area, and Segment Area (Radians)
1. Arc Length
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:
- \( r \) = radius of the circle
- \( \theta \) = angle in radians
✅ This formula comes directly from the definition of radians.
For a full revolution, \( \theta = 2\pi \), so \( l = 2\pi r \) (the circumference).
2. Area of a Sector
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 \( \theta / (2\pi) \).
3. Area of a Segment
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.
📝 Examples
Example 1: Find the arc length for \( r = 5 \) cm and \( \theta = \frac{\pi}{3} \) radians.
\[
l = r\theta = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} \ \text{cm}.
\]
Example 2: Find the area of a sector with \( r = 10 \) cm and \( \theta = 2 \) radians.
\[
A = \frac{1}{2} r^2 \theta
= \frac{1}{2} \times 10^2 \times 2
= 100 \ \text{cm}^2.
\]
Example 3: Find the area of a segment when \( r = 6 \) cm and \( \theta = 1.2 \) radians.
\[
\text{Segment area}
= \frac{1}{2} r^2 (\theta - \sin\theta)
= \frac{1}{2} \times 6^2 \times (1.2 - \sin 1.2)
\]
\[
= 18 \times (1.2 - 0.9320)
= 18 \times 0.268
\approx 4.82 \ \text{cm}^2.
\]
📌 Key Points:
- Always ensure the angle \(\theta\) is in radians when applying these formulas.
- Arc length is linear — it scales directly with \( \theta \).
- Sector area is proportional to \( \theta \) as a fraction of the full circle area \( \pi r^2 \).
- Segment area = sector area – triangle area formed by the two radii and chord.