Homeclass 10MathsAreas Related to CirclesMajor Segment

Major Segment

A major segment of a circle is the larger region enclosed by a chord and the corresponding major arc. Unlike the minor segment, the major segment always contains the center of the circle and represents the greater portion of the circle's area.

Key Concepts

A circle is divided into two unequal segments by any chord that is not a diameter; the larger of these two regions is the major segment.

The boundary of a major segment consists of a straight line segment (the chord) and a curved line (the major arc).

The area of a major segment is mathematically determined by subtracting the area of the corresponding minor segment from the total area of the circle.

Formula & Equation

\text{Area of Major Segment} = \pi r^2 - \left( \frac{\theta}{360^\circ} \times \pi r^2 - \frac{1}{2} r^2 \sin \theta \right)

r \text{ is the radius of the circle, and } \theta \text{ is the central angle subtended by the chord in degrees.}

Common Misconceptions

Myth: A major segment and a major sector are the exact same geometric shape.

Fact: A major segment is bounded by a chord and a major arc, while a major sector is bounded by two radii and a major arc. The major segment's area includes the triangle formed by the radii and the chord.

Real World Applications

The cross-section of water in a horizontal cylindrical pipe when the pipe is more than half full forms a major segment.

If you cut a small straight slice off the edge of a circular pizza, the larger remaining portion of the pizza represents a major segment.

Frequently Asked Questions

How do you calculate the area of a major segment?

You calculate the total area of the circle and subtract the area of the minor segment from it.

Does the major segment contain the center of the circle?

Yes, the major segment always contains the center of the circle, which is a key property that distinguishes it from the minor segment.

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