NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 11: Area Related To Circles - Exercise 11.2

Question 4

Step 1: Understand the terms.

• A sector of a circle looks like a slice of pizza. Its area is given by:
  \( A_{\text{sector}} = \tfrac{1}{2} r^2 \theta \), where:
  • \(r\) = radius (in metres)
  • \(\theta\) = angle at the centre (in radians)
• A segment is part of the circle cut off by a chord (a straight line inside the circle). It is smaller than the whole sector because it does not include the triangular part.

Step 2: Relation between sector and segment.

Area of segment = Area of sector − Area of triangle (formed by the two radii and the chord).

Step 3: Area of the triangle.

The triangle is isosceles, and its area is:

\( A_{\text{triangle}} = \tfrac{1}{2} r^2 \sin \theta \), which is always positive (for \(0 < \theta < 2\pi\)).

Step 4: Substitution.

So, area of the segment:

\( A_{\text{segment}} = \tfrac{1}{2} r^2 \theta - \tfrac{1}{2} r^2 \sin \theta \).

Step 5: Compare.

Since we subtract a positive quantity (triangle area), the segment’s area is always smaller than the sector’s area.

Final Answer: Yes, it is true. The area of a segment is less than the area of its corresponding sector.