Arcs and Sectors Flashcard Review

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

The length of an arc is calculated using the formula: $$s = r \theta$$ , where $$s$$ is the arc length, $$r$$ is the radius, and $$\theta$$ is the central angle in radians.

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

The radian measure of a 90-degree angle is $$\frac{\text{π}}{2}$$ radians.

The area of a circle is calculated using the formula: $$A = \text{π} r^2$$ . For a radius of 5 inches, the area is $$A = \text{π} (5)^2 = 25\text{π} \text{ in}^2$$ .

The area of a shaded sector can be calculated using: $$A = \frac{1}{2} r^2 \theta$$ , where $$\theta$$ is in radians.

The radius can be found using the formula: $$r = \frac{s}{\theta}$$ , where $$s$$ is the arc length and $$\theta$$ is the central angle in radians.

First, convert 60 degrees to radians: $$\frac{60 \times \text{π}}{180} = \frac{\text{π}}{3}$$ . Then, use the area formula: $$A = \frac{1}{2} (10)^2 \frac{\text{π}}{3} = \frac{50\text{π}}{3} \text{ in}^2$$ .

In the equation $$s = \theta r$$ , $$\theta$$ represents the central angle measured in radians.

Convert 45 degrees to radians: $$\frac{45 \times \text{π}}{180} = \frac{\text{π}}{4}$$ . Then, use the formula: $$r = \frac{s}{\theta} = \frac{15.8\text{π}}{\frac{\text{π}}{4}} = 63.2 \text{ ft}$$ .