Arc length and sector area

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

To convert degrees to radians, use the formula: $$radians = degrees \times \frac{\pi}{180}$$ .

The length of an arc is given by the formula: $$L = r \theta$$ , where $$r$$ is the radius and $$\theta$$ is the angle in radians.

First, convert 150 degrees to radians: $$150 \times \frac{\pi}{180} = \frac{5\pi}{6}$$ . Then, use the arc length formula: $$L = 6 \times \frac{5\pi}{6} = 5\pi$$ .

Convert 315 degrees to radians: $$315 \times \frac{\pi}{180} = \frac{7\pi}{4}$$ . Then, use the area formula: $$A = \frac{1}{2} \times 16^2 \times \frac{7\pi}{4} = 224\pi$$ .

The area of a sector increases with the square of the radius, as seen in the formula: $$A = \frac{1}{2} r^2 \theta$$ .

The angle determines the proportion of the circle that the arc represents; larger angles yield longer arcs.

Convert the angle from degrees to radians and then apply the arc length formula: $$L = r \theta$$ .

Convert 165 degrees to radians: $$165 \times \frac{\pi}{180} = \frac{11\pi}{12}$$ . Then, use the area formula: $$A = \frac{1}{2} \times 24^2 \times \frac{11\pi}{12} = 132\pi$$ .

The area of a circle is given by the formula: $$A = \pi r^2$$ .