Question

A chord of a circle, of radius 14 cm, subtends an angle of \(60^{\circ}\) at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

Hint:

If the central angle is \(60^{\circ}\), the triangle formed by the radii and the chord is always equilateral. If the angle is \(90^{\circ}\), use Pythagoras theorem to find the chord length.

Answer 1

Step 1: Understanding the Given Data:
Radius r = 14 cm
Central angle θ = 60°

We calculate:
1) Area of smaller sector
2) Perimeter of smaller segment

Step 2: Area of Sector (Alternative Simplification Method):
Area of full circle = πr²
= (22/7) × 14 × 14
= (22/7) × 196
= 22 × 28
= 616 cm²

Since 60° is 1/6 of full circle,
Area of sector = 616 / 6
= 308 / 3
≈ 102.67 cm²

Step 3: Length of Arc:
Circumference of circle = 2πr
= 2 × (22/7) × 14
= 88 cm

Arc length for 60° = 88 / 6
= 44 / 3
≈ 14.67 cm

Step 4: Length of Chord AB:
In triangle OAB:
OA = OB = 14 cm
∠AOB = 60°

Using cosine rule:
AB² = 14² + 14² − 2(14)(14)cos60°
= 196 + 196 − 392 × (1/2)
= 392 − 196
= 196

AB = 14 cm

Step 5: Perimeter of Smaller Segment:
Perimeter = Arc length + Chord length
= 44/3 + 14
= (44 + 42)/3
= 86/3
≈ 28.67 cm

Final Answer:
Area of smaller sector = 308/3 cm² (≈ 102.67 cm²)
Perimeter of smaller segment = 86/3 cm (≈ 28.67 cm)