Arcs and Central Angles
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what arcs and central angles are, how they relate to each other, and how to find their measures in a circle.
The Concept
Section titled “The Concept”A central angle is an angle whose vertex is at the center of the circle and whose sides pass through two points on the circle.
An arc is the curved portion of the circle between those two points.
Key relationships:
- The measure of a minor arc (the shorter arc) is equal to the measure of its central angle.
- The measure of a major arc (the longer arc) is 360° minus the measure of the minor arc.
- The entire circle is 360°.
Worked Example
Section titled “Worked Example”-
A central angle measures 90°. What is the measure of the minor arc?
90° (same as the central angle).
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A central angle is 120°. What is the measure of the major arc?
360° − 120° = 240°.
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The minor arc between points A and B measures 75°. What is the central angle?
75° (they are equal).
Real-World Application
Section titled “Real-World Application”Arcs and central angles are used in many practical situations:
- Cutting a pizza or cake into equal slices (each slice is defined by a central angle and its arc).
- Designing circular gardens, fountains, or patios with specific curved sections.
- Navigation: GPS and maps often describe turns as arcs with central angles.
- Sports: The path of a thrown ball or the curve of a racetrack.
- Architecture: Arched doorways, windows, or bridges use arc measurements.
For example, if you want to divide a circular flower bed into 5 equal sections, each central angle would be 360° ÷ 5 = 72°, and each arc would also measure 72°.