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

1. In a circle with a radius of \(5 \text{ cm}\), what is the length of the arc intercepted by a central angle of \(45^\circ\)?
2. 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?

1. \( \frac{45}{360} \times (2\pi \times 5) \approx 3.93 \text{ cm}\)
2. \( \left( \frac{20}{16\pi} \right) \times 360 \approx 143.31^\circ \)

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