CG Board Class 9 Mathematics Chapter 14 Solutions: Sector of A Circle and Length of Arc (वृत्त का त्रिज्यखण्ड एवं चाप की लम्बाई) – 2026-27

1. Select circular objects such as a coin, ring, or plate.
2. Measure the circumference (C) using a thread and the diameter (d) using a ruler.
3. Calculate the ratio C / d for each object.
4. Result: The ratio of circumference to diameter (C / d) is constant for every circle.
5. Value: This constant ratio is represented by the Greek symbol π (pi), with an approximate value of 22/7 or 3.14.

Given:

Radius of circle (r) = 17.5 cm

Formula:

Circumference of circle = 2 × π × r

Calculation:

Circumference = 2 × (22/7) × 17.5

= 2 × 22 × 2.5

= 110 cm

Given:

Radius of circle (r) = 4.2 cm

Formula:

Area of circle = π × r²

Calculation:

Area = (22/7) × (4.2)²

= (22/7) × 4.2 × 4.2

= 22 × 0.6 × 4.2

= 55.44 sq cm

Given:

Length of rope = Radius of grazing circle (r) = 14 m

Formula:

Grazing Area = Area of circle = π × r²

Calculation:

Area = (22/7) × (14)²

= (22/7) × 14 × 14

= 22 × 2 × 14

= 616 sq m

Given:

Radius of bicycle wheel (r) = 35 cm = 0.35 m

Number of rotations (n) = 500

Formulas:

Distance in 1 rotation = Circumference of wheel = 2 × π × r

Total distance = Number of rotations × Distance in 1 rotation

Calculation:

Circumference = 2 × (22/7) × 35 = 220 cm = 2.2 m

Total Distance = 500 × 220 cm = 110,000 cm = 1100 m

Given:

Radius of first circle (r₁) = 3 m

Area of first circle (A₁) = π × (r₁)² = π × 3² = 9π sq m

Let radius of second circle be R.

Area of second circle (A₂) = π × R²

According to question:

A₂ = 9 × A₁

π × R² = 9 × (9π)

R² = 81

R = √81 = 9 m

Given:

Inner circumference = 440 m

Width of path (w) = 14 m

Inner Radius (r₂) Calculation:

2 × π × r₂ = 440

2 × (22/7) × r₂ = 440

(44/7) × r₂ = 440

r₂ = (440 × 7) / 44 = 70 m

Outer Radius (r₁) Calculation:

r₁ = r₂ + w = 70 + 14 = 84 m

Outer Diameter (D₁) Calculation:

Outer Diameter = 2 × r₁ = 2 × 84 = 168 m

Given:

Inner radius (ground radius, r₂) = 50 m

Width of road = 5 m

Outer radius (r₁) = 50 + 5 = 55 m

Rate of tiling = ₹ 30 per sq m

Formula:

Area of road = π × (r₁² – r₂²)

Calculation:

Area of road = (22/7) × (55² – 50²)

= (22/7) × (3025 – 2500)

= (22/7) × 525

= 22 × 75 = 1650 sq m

Total Cost = Area of road × Rate per sq m

Total Cost = 1650 × 30 = ₹ 49,500

Given:

Radius of circle (r) = 21 cm

Angle subtended at centre (θ) = 70°

Formulas:

Length of arc = (θ / 360°) × 2 × π × r

Area of sector = (θ / 360°) × π × r²

Calculation:

1. Length of Arc:

Length of arc = (70° / 360°) × 2 × (22/7) × 21

= (7 / 36) × 132

= 924 / 36 = 25.66 cm (or 25.6 cm)

1. Area of Sector:

Area of sector = (70° / 360°) × (22/7) × (21)²

= (7 / 36) × (22/7) × 441

= (1 / 36) × 22 × 441

= 9702 / 36 = 269.5 sq cm

Given:

Area of sector = (1 / 6) × Area of circle

Formula:

Area of sector = (θ / 360°) × Area of circle

Calculation:

(θ / 360°) × Area of circle = (1 / 6) × Area of circle

θ / 360° = 1 / 6

θ = 360° / 6 = 60°

Given:

Area of sector = 1540 sq cm

Angle at centre (θ) = 50°

Formula:

Area of sector = (θ / 360°) × π × r²

Calculation:

1540 = (50° / 360°) × (22/7) × r²

1540 = (5 / 36) × (22/7) × r²

1540 = (110 / 252) × r²

r² = (1540 × 252) / 110

r² = 14 × 252

r² = 3528

r = √3528 = √(1764 × 2) = 42√2 cm