What is the Relationship Between Arcs and Central Angles?
A central angle has its vertex at the center of a circle, and it equals the measure of the arc it intercepts exactly. That one-to-one relationship is the foundation for every other arc and angle rule.
- Arc: An arc is a segment or a portion of the circumference of a circle.
- Central Angle: It is an angle whose vertex is at the center of the circle and whose sides intercept an arc on the circle.
Worked examples: arcs and central angles
Practice questions on arcs and central angles
- In a circle with a radius of (5 text{ cm}), what is the length of the arc intercepted by a central angle of (45^circ)?
- A circle has a central angle that intercepts an arc of (20 text{ cm}) in length. If the circle’s radius is (8 text{ cm}), what is the measure of the central angle?
- ( frac{45}{360} times (2pi times 5) approx 3.93 text{ cm})
- ( left( frac{20}{16pi} right) times 360 approx 143.31^circ )
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