Central Angles & Arc Measure
A central angle has its vertex at the center of the circle. Its measure equals the measure of the intercepted arc — the simplest angle-arc relationship.
Introduction
Central angle θ = intercepted arc AB
Past Knowledge
Circle vocabulary (10.1.1). Angle measurement. Protractors.
Today's Goal
Relate central angles to arc measure; find missing arcs.
Future Success
Arc addition (10.1.3), inscribed angles (10.1.4), arc length (11.2.2).
Key Concepts
Central Angle Theorem
If , then
Key Relationships
- Minor arc + Major arc = 360°
- In congruent circles, congruent central angles → congruent arcs → congruent chords
- A diameter creates two semicircles of 180° each
Worked Examples
Central Angle → Arc
Central angle ∠AOB = 130°. Find arc AB and arc ACB (major).
Minor arc AB = 130° (equals central angle)
Major arc ACB = 360° − 130° = 230°
arc AB = 130°, arc ACB = 230°
Finding a Central Angle
Circle with center O. Arc AB = 85°, arc BC = 120°, arc CD = 95°. Find arc DA and ∠DOA.
(central angle = arc)
arc DA = ∠DOA = 60°
Algebraic
Three consecutive arcs measure (3x)°, (5x + 10)°, and (2x + 30)°. They make up the full circle. Find each arc.
→
Arcs: 96°, 170°, 94°. Check: 96 + 170 + 94 = 360 ✓
96°, 170°, 94°
Common Pitfalls
Arc Measure ≠ Arc Length
Arc measure is in degrees (how much of the 360° circle). Arc length is in linear units (actual distance along the curve). We'll cover length in 11.2.2.
Central Angle Only When Vertex Is at Center
The “central angle = arc” rule ONLY works when the vertex is literally the center. An inscribed angle (vertex on the circle) follows a different rule.
Real-Life Applications
Clocks
Each hour mark on a clock is separated by a 30° central angle (360° ÷ 12). At 3:00, the hands form a 90° central angle intercepting a 90° arc.
Pie Charts
Each slice of a pie chart is a sector. If a category is 25% of data, its central angle is 0.25 × 360° = 90°.
Practice Quiz
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