📚 Topic Summary
The arc length of a circle is the distance along the curved line making up the arc. It can be found using the formula $s = r\theta$, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians. The sector area is the area enclosed by an arc and two radii. It can be found using the formula $A = \frac{1}{2}r^2\theta$, where $A$ is the sector area, $r$ is the radius, and $\theta$ is the central angle in radians.
Remember to always convert the central angle to radians before applying these formulas. Radians are calculated using the conversion factor $\frac{\pi}{180}$ from degrees to radians. Let's practice these concepts!
🔤 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Arc Length |
A. The area enclosed by an arc and two radii of a circle. |
| 2. Sector Area |
B. An angle whose vertex is the center of a circle. |
| 3. Radian |
C. The distance along the curved line making up an arc. |
| 4. Central Angle |
D. A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius. |
| 5. Radius |
E. The distance from the center of a circle to any point on its circumference. |
Match the numbers to the letters for the correct answer.
✍️ Part B: Fill in the Blanks
Complete the following sentences using the words provided: radius, arc length, sector area, radians, central angle.
- To calculate the __________, you need to know the radius and the central angle in __________.
- The __________ is the distance from the center of the circle to any point on it.
- The __________ is a portion of the circumference of a circle.
🤔 Part C: Critical Thinking
Explain how the formulas for arc length and sector area are related. Can you derive one formula from the other?