Arc Length

Arc length in a circle can be calculated using the formula $s = r\theta$, where $s$ represents the arc length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians.

Image courtesy to Cuemath

Example 1:

Calculate the arc length subtended by an angle of $\frac{\pi}{4}$radians in a circle with a radius of 4 units.

Solution:

Arc length $= 4 \times \frac{\pi}{4} = \pi$ units.

Example 2:

Determine the arc length when the central angle is $\frac{\pi}{2}$radians and the radius is 3 units.

Solution:

Arc length $= 3 \times \frac{\pi}{2} = \frac{3\pi}{2}$units.

Area of a Sector

The area of a sector is given by $A = \frac{1}{2} r^2 \theta$, where $A$ is the area, $r$ is the radius, and $\theta$ is the central angle in radians.

Image courtesy to Cuemath

Example 1:

Calculate the area of a sector where $r = 5$ units and $\theta = \frac{\pi}{3}$ radians.

Solution:

Area $= \frac{1}{2} \times 5^2 \times \frac{\pi}{3} = \frac{25\pi}{6}$ square units.

Example 2:

Find the area of a sector with a radius of 7 units and a central angle of $\frac{2\pi}{3}$ radians.

Solution:

Area $= \frac{1}{2} \times 7^2 \times \frac{2\pi}{3} = \frac{49\pi}{3}$ square units.