# How to Determine Segment Measures in Circles

Circles, one of the most fundamental shapes in geometry, captivate mathematicians and scholars with their simplicity and complexity alike. Among the various aspects of circle geometry, segment measures stand out as a crucial area of study. These measures refer to the lengths or areas of the portions of the circle delineated by chords, arcs, and sometimes tangents. The precise determination of these segment measures not only unveils the inherent relationships within the circle but also forms the bedrock for advanced mathematical explorations. As we venture into this topic, we will encounter principles that govern the measure of these segments, enriching our understanding of the interplay between different elements within the circle.

## Step-by-step Guide: Segment Measures

**Definitions**:

**Arc**: A continuous piece of the circle.**Arc Measure**: The degree measure of an arc, which is the same as the central angle intercepting that arc.**Segment**: A region in a circle bounded by a chord and the arc subtended by the chord.**Segment Measure**: The area of a segment can be found by subtracting the area of the sector from the area of the triangle formed by the chord and the radii connecting the chord’s endpoints to the center.

**Formula for Segment Measure (Area)**:

Segment area \( = \) Area of sector \( – \) Area of triangle

Segment area \( = \frac{1}{2} r^2 \theta – \frac{1}{2} r^2 \sin(\theta) \)

Where:

- \( r \) is the radius of the circle.
- \( \theta \) is the measure of the central angle in radians.

### Examples

**Example 1**:

Determine the area of a segment in a circle of radius \(10 \text{ cm}\) with a central angle of \(\pi/3\) radians.

**Solution:**

Using the formula:

Segment area \( = \frac{1}{2} r^2 \theta – \frac{1}{2} r^2 \sin(\theta) \)

Plugging in the values, we get:

Segment area \( = \frac{1}{2} (10^2) \frac{\pi}{3} – \frac{1}{2} (10^2) \sin(\pi/3) \)

Segment area \( = \frac{1}{2} \times 100 \times \frac{\pi}{3} – \frac{1}{2} \times 100 \times \frac{\sqrt{3}}{2} \)

Segment area \( \approx 9.06 \text{ cm}^2 \)

The area of the segment is approximately \(9.06 \text{ cm}^2\).

**Example 2:**

For a circle with radius \(7 \text{ cm}\) and a central angle of \(\pi/4\) radians, find the area of the segment.

**Solution:**

Using the formula:

Segment area \( = \frac{1}{2} r^2 \theta – \frac{1}{2} r^2 \sin(\theta) \)

Plugging in the values:

Segment area \( = \frac{1}{2} (7^2) \frac{\pi}{4} – \frac{1}{2} (7^2) \sin(\pi/4) \)

Segment area \( = \frac{1}{2} \times 49 \times \frac{\pi}{4} – \frac{1}{2} \times 49 \times \frac{\sqrt{2}}{2} \)

Segment area \( \approx 1.9 \text{ cm}^2 \)

The area of the segment is approximately \(1.9 \text{ cm}^2\).

**Practice Questions**:

- Calculate the area of a segment for a circle with radius \(5 \text{ cm}\) and a central angle of \(\pi/6\) radians.
- Determine the segment area of a circle with a \(12 \text{ cm}\) radius and a central angle of \(\pi/2\) radians.

**Answers**:

- \( \approx 0.29 \text{ cm}^2 \)
- \( \approx 41.04 \text{ cm}^2 \)

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