Central Angles and Arcs - Key Concepts
Understanding Central Angles
- Definition: A central angle is an angle whose vertex is the center of the circle and whose sides are radii of the circle.
- Measurement: Central angles are usually measured in degrees.
Relationship Between Central Angles and Arcs
- Arc: Part of the circumference of a circle.
- Arc Length: The distance along the curved path of the circle segment.
- Central Angle Relation: The measure of a central angle is equal to the measure of its intercepted arc.
Calculating Arcs
- Arc Measure: Equal to the measure of the central angle that intercepts it.
- Arc Length Formula:
[ \text{Arc Length} = 2\pi r \times \left( \frac{\text{Central Angle}}{360} \right) ]
Where ( r ) is the radius of the circle.
- Example: If a central angle measures 60 degrees in a circle with a radius of 5, its arc length is ( \frac{1}{6} ) of the total circumference.
Types of Arcs
- Minor Arc: An arc that is smaller than a semicircle (less than 180 degrees).
- Major Arc: An arc that is larger than a semicircle (more than 180 degrees).
- Semicircle: An arc that is exactly half of the circle, measuring 180 degrees.
Practice Problem
- Problem Statement: Calculate the arc length of a central angle measuring 45 degrees in a circle with a radius of 10.
- Solution: Using the arc length formula: ( 2\pi \times 10 \times \frac{45}{360} = 2.5\pi ) units.
Additional Points
- Sector: The region bounded by two radii and the arc.
- Sector Area Formula:
[ \text{Sector Area} = \pi r^2 \times \left( \frac{\text{Central Angle}}{360} \right) ]
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