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Area Of A Segment

The area of a segment of a circle is the region enclosed between a chord and its corresponding arc. It is calculated by finding the area of the sector formed by the chord's endpoints and subtracting the area of the triangle formed by the radii and the chord.

Key Concepts

A chord divides a circle into two distinct regions: the minor segment, which is smaller, and the major segment, which is larger.
To calculate the area of a minor segment, you must first determine the area of the corresponding sector and then subtract the area of the triangle formed by the central angle.
The area of a major segment can be found by calculating the total area of the circle and subtracting the area of the minor segment.

Formula & Equation

θ360∘×πr2−12r2sin⁡θ\frac{\theta}{360^\circ} \times \pi r^2 - \frac{1}{2} r^2 \sin\theta

r represents the radius of the circle, and \theta represents the central angle subtended by the chord in degrees.

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Common Misconceptions

Myth: Students often confuse a segment with a sector and use the sector formula to find the area of a segment.
Fact: A sector is shaped like a slice of pizza bounded by two radii and an arc, whereas a segment is bounded by a straight chord and an arc. You must subtract the triangle's area from the sector's area to find the segment's area.

Real World Applications

Calculating the cross-sectional area of water flowing through a partially filled horizontal cylindrical pipe.
Designing architectural elements like arched doorways or stained glass windows where a straight wooden panel cuts across a circular frame.

Frequently Asked Questions

How do you find the area of a segment of a circle?
First, calculate the area of the sector using the central angle and radius. Then, subtract the area of the triangle formed by the two radii and the chord.
What is the difference between a minor segment and a major segment?
A minor segment is the smaller region bounded by a chord and a minor arc, while the major segment is the larger region bounded by the same chord and the major arc.

Keep Learning

Segment of a CircleArea of a SectorArea of a Circle