Is it true to say that area of segment of a circle is less than the area of its corresponding sector? Why ?

Explanation

To determine whether the area of a segment of a circle is less than the area of its corresponding sector, we can analyze the two cases: the major segment and the minor segment.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- A **sector** of a circle is a region enclosed by two radii and the arc between them.
- A **segment** of a circle is a region enclosed by a chord and the arc that subtends it.

2. **Case 1: Major Segment**:
- Consider a major segment formed by a chord \( AB \) and the arc \( ACB \).
- The area of the major segment \( ACB \) includes the area of the triangle \( AOB \) (where \( O \) is the center of the circle).
- The area of the corresponding sector \( AOB \) is the area enclosed by the radii \( OA \), \( OB \), and the arc \( AB \).

**Comparison**:
- The area of the major segment \( ACB \) is greater than the area of the sector \( AOB \) because the area of triangle \( AOB \) is included in the major segment.

3. **Case 2: Minor Segment**:
- Now consider a minor segment formed by the same chord \( AB \) and the arc \( ACB \).
- The area of the minor segment \( ACB \) is less than the area of the corresponding sector \( AOB \) because the area of triangle \( AOB \) is included in the sector.

**Comparison**:
- The area of the sector \( AOB \) is greater than the area of the minor segment \( ACB \).

4. **Conclusion**:
- The statement that the area of a segment of a circle is less than the area of its corresponding sector is **true only for the minor segment**.
- For the major segment, the area of the segment is greater than the area of the corresponding sector.

### Final Statement:
Thus, it is not universally true that the area of a segment of a circle is less than the area of its corresponding sector; it depends on whether the segment is a minor or major segment.