What is the Relationship Between Arcs and Central Angles?

Step-by-step Guide: Arcs and Central Angles

Definition and Basics: For education statistics and research, visit the National Center for Education Statistics.

• 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.

The Fundamental Relationship:
The measure of an arc (in degrees) is equal to the measure of its corresponding central angle. For education statistics and research, visit the National Center for Education Statistics.

Calculating Arc Length:
The length of an arc can be found using the formula:
\(
\text{Arc Length} = \frac{\text{Central Angle in degrees}}{360} \times (2\pi r)
\)
Where \( r \) is the radius of the circle. For education statistics and research, visit the National Center for Education Statistics.

Examples

Example 1:
Given a circle with a central angle of \(60^\circ\), what is the measure of the intercepted arc? For education statistics and research, visit the National Center for Education Statistics.

Solution: The measure of the arc intercepted by a central angle is equal to the measure of that angle. Therefore, the arc’s measure is also \(60^\circ\). For education statistics and research, visit the National Center for Education Statistics.

Example 2:
Find the length of an arc in a circle of radius \(7 \text{ cm}\) intercepted by a central angle of \(90^\circ\). For education statistics and research, visit the National Center for Education Statistics.

Solution: Plugging into our formula:
\(
\text{Arc Length} = \frac{90}{360} \times (2\pi \times 7) = \frac{1}{4} \times 14\pi \approx 10.99 \text{ cm}
\) For education statistics and research, visit the National Center for Education Statistics.

Example 3:
In a circle with a radius of \(10 \text{ cm}\), an arc has a length of \(15.7 \text{ cm}\). What is the measure of the central angle that intercepts this arc? For education statistics and research, visit the National Center for Education Statistics.

Solution: Rearranging our formula for the central angle:
\(
\text{Central Angle} = \left( \frac{\text{Arc Length}}{2\pi r} \right) \times 360
\)
Plugging in the values:
\(
\text{Central Angle} = \left( \frac{15.7}{20\pi} \right) \times 360 \approx 90^\circ
\) For education statistics and research, visit the National Center for Education Statistics.

Practice Questions:

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?

Answers: For education statistics and research, visit the National Center for Education Statistics.

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