NCERT Class 10 Maths · Chapter 11
NCERT Solutions Class 10 Maths Chapter 11 — Areas Related to Circles
Step-by-step solutions for all exercises in NCERT Class 10 Maths Areas Related to Circles.
Chapter Overview
Calculate area of sectors, segments, and combinations of plane figures involving circles.
This chapter is part of the NCERT Mathematics textbook for Class 10 and is important for CBSE school examinations. The concepts covered here build the foundation for more advanced topics in higher classes.
Below you will find solved examples from this chapter. Each solution includes detailed step-by-step working so you can understand the method, not just the answer.
Solved Examples from Areas Related to Circles
1Which of the following statements correctly defines a 'sector' of a circle?
Answer: The region enclosed by two radii and their corresponding arc.
Solution:
Step 1: A sector of a circle is formed by two radii originating from the center of the circle and the arc connecting their endpoints on the circle's circumference.
Step 2: Option A describes a segment, not a sector. Option B is incorrect as it mentions a diameter, which is not a defining element for a sector. Option D is also incorrect.
Step 3: Thus, the correct definition involves two radii and the arc.
2A student calculates the area of a sector with radius 'r' and angle 'θ' as (θ/360°) × πr². Their friend says this formula is only for minor sectors. Is the friend correct?
Answer: No, the formula applies to both minor and major sectors.
Solution:
Step 1: The formula for the area of a sector, Area = (θ/360°) × πr², is a general formula.
Step 2: If 'θ' represents the angle of the minor sector (e.g., 60°), it calculates the minor sector's area.
Step 3: If 'θ' represents the angle of the major sector (i.e., 360° - angle of minor sector, e.g., 300°), it calculates the major sector's area.
Step 4: Therefore, the formula is universally applicable, provided the correct angle 'θ' for the desired sector is used.
3Ravi wants to find the length of the arc of a sector with radius 7 cm and angle 90°. He applies the formula: Arc Length = (90/360) × π(7)². What mistake did Ravi make?
Answer: He used the formula for the area of a sector instead of arc length.
Solution:
Step 1: The formula for the length of an arc is (θ/360°) × 2πr.
Step 2: Ravi used (θ/360°) × πr², which is the formula for the area of a sector, not the length of an arc.
Step 3: The angle (90°) and radius (7 cm) are correctly identified for the problem, but the base formula applied is wrong.
Step 4: Therefore, the mistake is in using the area formula instead of the arc length formula.
4What is the area of a quadrant of a circle with a radius of 14 cm? (Use π = 22/7)
Answer: 154 cm²
Solution:
Step 1: A quadrant of a circle is a sector with a central angle of 90°.
Step 2: The area of a circle is given by A = πr².
Step 3: The area of a quadrant is (1/4) × πr² or (90/360) × πr².
Step 4: Substituting r = 14 cm and π = 22/7: Area = (1/4) × (22/7) × (14)² = (1/4) × (22/7) × 196 = (1/4) × 22 × 28 = 22 × 7 = 154 cm².
5The area of a segment of a circle is calculated by subtracting the area of the corresponding __________ from the area of the corresponding __________.
Answer: triangle, sector
Solution:
Step 1: A segment of a circle is the region bounded by an arc and its corresponding chord.
Step 2: To find the area of this segment, we first consider the sector formed by the same arc and the two radii to its endpoints.
Step 3: From this sector, we subtract the area of the triangle formed by the two radii and the chord.
Step 4: Therefore, the area of a segment = Area of sector - Area of corresponding triangle.
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