Homeclass 10MathsAreas Related to CirclesMinor Segment

Minor Segment

A minor segment of a circle is the smaller region enclosed by a chord and its corresponding minor arc. When a chord divides a circular region into two unequal parts, the part with the smaller area is called the minor segment.

Key Concepts

It is bounded by a straight line segment known as the chord and a curved line known as the minor arc.

The central angle subtended by the minor arc of a minor segment is always strictly less than 180 degrees.

To calculate its area, you must first find the area of the corresponding minor sector and then subtract the area of the triangle formed by the chord and the two radii connecting to its endpoints.

Formula & Equation

\text{Area of Minor Segment} = \frac{\theta}{360^\circ} \times \pi r^2 - \frac{1}{2} r^2 \sin\theta

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

Common Misconceptions

Myth: Students often confuse a minor segment with a minor sector.

Fact: A minor sector is shaped like a slice of pizza bounded by two radii and an arc, while a minor segment is bounded by a single chord and an arc.

Real World Applications

A straight cut made near the edge of a round cake or pizza, leaving a small curved piece.

The cross-section of water flowing at the bottom of a partially filled horizontal cylindrical pipe.

Frequently Asked Questions

How do you calculate the area of a minor segment?

You calculate it by subtracting the area of the triangle formed by the chord and radii from the total area of the corresponding minor sector.

What happens to the minor segment if the chord becomes the diameter?

If the chord is the diameter, it divides the circle into two equal parts, and the segment becomes a semicircle rather than a minor segment.

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