# What is the Relationship Between Arcs and Central Angles?

The circle is one of the most universally recognized shapes, yet within it lies a world of intricate relationships and properties. One such fascinating aspect is the connection between arcs and central angles. While these terms might sound complex, understanding them is key to unlocking many wonders of circular geometry. Join us as we delve into a step-by-step exploration of arcs and their relationship with central angles, illustrating how these concepts interplay to define the geometry of circles.

## 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:

- 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?

**Answers:**

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

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