Areas Related to Circles – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

This chapter finds the area and perimeter of sectors and segments of a circle and of figures made from circles.

Sector

• Area of a sector of angle θ = (θ/360°) × πr².
• Length of the arc = (θ/360°) × 2πr.
• Perimeter of a sector = 2r + arc length.

Segment

• Area of minor segment = area of sector − area of the triangle formed by the radii and chord.
• Area of major segment = πr² − area of minor segment.
• For θ = 60°, the triangle is equilateral: area (√3/4)r².

Combination figures

• Add or subtract areas of the simple shapes.
• A minute hand sweeps 360° in 60 minutes (6° per minute).

Key terms

Sector

region between two radii and an arc

Segment

region between a chord and an arc

Minor arc

the shorter arc between two points

Formulas

• Area of sector = (θ/360) × πr²
• Arc length = (θ/360) × 2πr
• Area of segment = (θ/360)πr² − ½r² sin θ

Important questions with answers

Quick revision

• Sector area = θ/360 of the circle.
• Segment = sector − triangle.
• Perimeter of sector = 2r + arc.

Exam tips

• Use π = 22/7 when the radius is a multiple of 7, else 3.14.
• For clock questions convert minutes to degrees first.

Practice MCQs

1. Area of a sector of angle 90° in a circle of radius 14 cm is

   1. 616 cm²
   2. 154 cm²
   3. 308 cm²
   4. 77 cm²
   Show answer

   Answer: (b) 154 cm². ¼ × 22/7 × 196 = 154 cm².

2. Arc length of a 60° sector of radius 21 cm is

   1. 11 cm
   2. 22 cm
   3. 44 cm
   4. 66 cm
   Show answer

   Answer: (b) 22 cm. (60/360) × 2 × 22/7 × 21 = 22 cm.

3. The angle a minute hand turns in 5 minutes is

   1. 30°
   2. 5°
   3. 60°
   4. 15°
   Show answer

   Answer: (a) 30°. 6° per minute × 5.

4. Area swept by a 14 cm minute hand in 5 minutes is

   1. 51.33 cm²
   2. 102.67 cm²
   3. 154 cm²
   4. 25.67 cm²
   Show answer

   Answer: (a) 51.33 cm². (30/360) × 22/7 × 196 = 51.33 cm².

5. Perimeter of a quadrant of radius 7 cm is

   1. 11 cm
   2. 25 cm
   3. 18 cm
   4. 32 cm
   Show answer

   Answer: (b) 25 cm. 2 × 7 + 11 = 25 cm.

6. Area of the minor segment for a 90° angle in a circle of radius 10 cm (π = 3.14) is

   1. 28.5 cm²
   2. 78.5 cm²
   3. 50 cm²
   4. 21.5 cm²
   Show answer

   Answer: (a) 28.5 cm². 78.5 − 50 = 28.5 cm².

7. If the perimeter and area of a circle are numerically equal, the radius is

   1. 1 unit
   2. 2 units
   3. 4 units
   4. π units
   Show answer

   Answer: (b) 2 units. 2πr = πr² ⇒ r = 2.

8. A horse is tied with a 5 m rope at a corner of a square field of side 15 m. The area it can graze (π = 3.14) is

   1. 39.25 m²
   2. 78.5 m²
   3. 19.625 m²
   4. 25 m²
   Show answer

   Answer: (c) 19.625 m². Quadrant: ¼ × 3.14 × 25 = 19.625 m².