Sectors (Leaving Cert Mathematics): Revision Notes

Arc Length s:

$s = r\theta$

Where:

• $s$ is the arc length,
• $r$ is the radius of the circle,
• $\theta$ is the angle in radians.

Area of a Sector A:

$A = \frac{1}{2}r^2\theta$

Where:

• $A$ is the area of the sector,
• $r$ is the radius,
• $\theta$ is the angle in radians.

Example 2: Find the arc length subtended by an angle of $\frac{\pi}{4}$ radians in a circle of radius $10$ cm.

• Solution: $s = r\theta = 10 \times \frac{\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2} \text{ cm}$

Example 3: Calculate the area of a sector with a central angle of $\frac{\pi}{3}$ radians and a radius of $6$ cm.

• Solution: $A = \frac{1}{2}r^2\theta = \frac{1}{2} \times 6^2 \times \frac{\pi}{3} = \frac{1}{2} \times 36 \times \frac{\pi}{3} = 18 \times \frac{\pi}{3} = 6\pi \text{ cm}^2$

Example: Find the area and arc length of the following sector:

• Arc Length:

• Area: