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 Concept:
A sector is a part of a circle bounded by two radii and an arc. A segment is bounded by a chord and an arc.
Step 2: Key Formula or Approach:
Area of Sector \(= \frac{\theta}{360} \times \pi r^{2}\)
Length of Arc \(= \frac{\theta}{360} \times 2\pi r\)
Perimeter of segment \(=\) Length of arc \(+\) Length of chord.
Step 3: Detailed Explanation:
Given: \(r = 14\) cm, \(\theta = 60^{\circ}\).
Area of Sector:
\[ \text{Area} = \frac{60}{360} \times \frac{22}{7} \times 14 \times 14 \]
\[ \text{Area} = \frac{1}{6} \times 22 \times 2 \times 14 = \frac{308}{3} \approx 102.67 \text{ cm}^{2} \]
Perimeter of Segment:
1. Length of arc \(AB = \frac{60}{360} \times 2 \times \frac{22}{7} \times 14 = \frac{1}{6} \times 44 \times 2 = \frac{44}{3} \approx 14.67\) cm.
2. Length of chord \(AB\): Since \(\theta = 60^{\circ}\) and \(OA = OB = 14\), \(\triangle OAB\) is equilateral.
Thus, chord \(AB = 14\) cm.
Perimeter \(= \text{Arc length} + \text{Chord length}\)
\[ \text{Perimeter} = 14.67 + 14 = 28.67 \text{ cm} \]
Step 4: Final Answer:
The area of the smaller sector is \(102.67 \text{ cm}^{2}\) and the perimeter of the smaller segment is 28.67 cm.