How to Determine Segment Measures in Circles

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.

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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:

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

Answers:

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