What is the Relationship Between Arcs and Central Angles?

Step-by-step Guide: Arcs and Central Angles

Definition and Basics:

• 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.

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.

Examples

Example 1:
Given a circle with a central angle of \(60^\circ\), what is the measure of the intercepted arc?

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 \(60^\circ\).

Example 2:
Find the length of an arc in a circle of radius \(7 \text{ cm}\) intercepted by a central angle of \(90^\circ\).

Solution: Plugging into our formula:
\(
\text{Arc Length} = \frac{90}{360} \times (2\pi \times 7) = \frac{1}{4} \times 14\pi \approx 10.99 \text{ cm}
\)

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?

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:
\(
\text{Central Angle} = \left( \frac{15.7}{20\pi} \right) \times 360 \approx 90^\circ
\)

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:

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