Area of a sector examples and solutions.

📚 Quick Study Guide

• 📐 Definition: A sector of a circle is a region bounded by two radii and an arc of the circle.
• 🧮 Area Formula (degrees): If $\theta$ is the central angle in degrees, the area $A$ of the sector is given by: $A = \frac{\theta}{360} \pi r^2$, where $r$ is the radius of the circle.
• 🔄 Area Formula (radians): If $\theta$ is the central angle in radians, the area $A$ of the sector is given by: $A = \frac{1}{2} r^2 \theta$, where $r$ is the radius of the circle.
• 🔑 Key Concept: The area of a sector is a fraction of the total area of the circle, determined by the central angle.

Practice Quiz

1. What is the area of a sector with a radius of 5 cm and a central angle of 90 degrees?
   1. 6.25π cm²
   2. 25π cm²
   3. 5π cm²
   4. 12.5π cm²
2. A sector has an area of 12π square inches and a radius of 6 inches. What is the measure of the central angle in radians?
   1. $\frac{2\pi}{3}$
   2. $\frac{\pi}{3}$
   3. $\frac{4\pi}{3}$
   4. $\frac{\pi}{6}$
3. What is the area of a sector with a radius of 8 cm and a central angle of 45 degrees?
   1. 4π cm²
   2. 8π cm²
   3. 16π cm²
   4. 2π cm²
4. A circle has a radius of 10 cm. A sector of the circle has a central angle of 60 degrees. What is the area of the sector?
   1. $\frac{50\pi}{3}$ cm²
   2. $\frac{25\pi}{3}$ cm²
   3. $\frac{100\pi}{3}$ cm²
   4. $25\pi$ cm²
5. The area of a sector is 15π square meters and the central angle is 120 degrees. What is the radius of the circle?
   1. 7.5 m
   2. $\sqrt{45}$ m
   3. 6 m
   4. $\sqrt{40}$ m
6. A sector has a radius of 4 inches and a central angle of $\frac{\pi}{2}$ radians. What is the area of the sector?
   1. 2π in²
   2. 4π in²
   3. 8π in²
   4. $\frac{\pi}{4}$ in²
7. A sector of a circle has an area of 9π cm² and a radius of 6 cm. What is the measure of the central angle in degrees?
   1. 45°
   2. 60°
   3. 90°
   4. 120°