Central Angles - Definition, Examples, Quiz, FAQ, Trivia
Learn how angles at the center of a circle relate to arcs, sectors, and other circle parts
What is a Central Angle?
A central angle is an angle that has its vertex at the center of a circle. The sides of the angle pass through two points on the circle's circumference, creating an arc between them.
Think of it like a slice of pizza! The tip of the pizza slice is at the center of the circle, and the crust is the arc.
Central angles are measured in degrees. A full circle is 360 degrees, so the central angle for an entire circle is 360°. For a semicircle (half circle), the central angle would be 180°.
Key Fact
The measure of a central angle is equal to the measure of the arc it creates.
Remember
Every circle has 360 degrees. If you know what fraction of the circle your arc represents, you can find the central angle!
Related Circle Concepts
Central angles help us understand other parts of a circle:
Arc
The curved part of the circle between two points. The central angle determines the arc's length.
Sector
A "pizza slice" shaped area bounded by two radii and an arc. The central angle determines the sector's size.
Chord
A straight line connecting two points on a circle. The central angle helps determine the chord's length.
Inscribed Angle
An angle with its vertex on the circle. It's half the measure of the central angle that creates the same arc.
Relationship
The inscribed angle is always half the measure of the central angle that creates the same arc!
Central Angle Theorem
The Central Angle Theorem states that the measure of a central angle is equal to the measure of its intercepted arc. This is why we can say:
Central Angle Formula
But only when the angle is measured in radians! For degrees, we use fractions of the circle.
Central Angle vs. Inscribed Angle: When two angles create the same arc, the central angle is always twice as large as any inscribed angle.
This means if you have an inscribed angle of 30 degrees, the central angle for the same arc would be 60 degrees.
Theorem Tip
In a circle, all central angles that create equal arcs are themselves equal!
Real-World Examples
Central angles are all around us! Here are some examples:
Pizza Slices: Each slice of pizza has a central angle at its tip. If a pizza is cut into 8 equal slices, each central angle is 360° ÷ 8 = 45°.
Clock Faces: At 3:00, the hour hand is at the 3 and the minute hand is at the 12. The central angle between them is 90° (one-fourth of the circle).
Ferris Wheel: The cabins are arranged around the center. The angle between two adjacent cabins is a central angle.
Dartboard: The scoring sections are divided by central angles from the center of the board.
Bicycle Wheel: The spokes form central angles between them at the hub (center) of the wheel.
Try This
Look at a clock right now. What's the central angle between the hour and minute hands?
Central Angle Quiz
Test your understanding with this 5-question quiz. Choose the correct answer for each question.
Frequently Asked Questions
Here are answers to common questions about central angles:
Circle Trivia
Discover interesting facts about circles and angles:
Ancient Circles
The ancient Egyptians used circles in their architecture over 4,500 years ago. The Great Pyramids show remarkable understanding of circular geometry in their construction.
Nature's Circles
Many natural formations are circular: tree rings, ripples in water, spider webs, and even planets and stars! Nature often uses circles because they're efficient shapes.
Pi Connection
The ratio of a circle's circumference to its diameter is always π (pi), approximately 3.14159. This constant appears in formulas for arcs and sectors related to central angles.
Circular Art
Famous artist Leonardo da Vinci studied circles intensely. His "Vitruvian Man" drawing perfectly combines circular and human geometry, showing ideal proportions.
