Question 4 of 7intermediate🔧 ApplyNumerical4 marks

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and 3\sqrt{3} = 1.73)

Correct Answer

Area of minor segment = 20.44 cm² (approx.) Area of major segment = 686.06 cm² (approx.)

Exercise: EXERCISE 11.1 | Q: 6 | (Page Page 5)
For More Understanding

Explanation

This question directly matches Example 6 from the textbook context. The solution requires finding the area of the minor segment first using the formula: Area of segment = Area of sector – Area of triangle. The major segment area is then found by subtracting the minor segment from the total area of the circle. Since the angle is 60°, the triangle formed by the two radii and chord is equilateral, allowing us to use the formula (3\sqrt{3}/4) × side² for the triangle's area.

Solution Steps

  1. Step 1: Calculate Area of Sector OAPB Area of sector = (θ/360) × πr² = (60/360) × 3.14×153.14 \times 15 × 15 = (1/6) × 3.14×2253.14 \times 225 = 117.75 cm²

  2. Step 2: Calculate Area of Triangle OAB Since the angle subtended is 60° and OA = OB = radius, triangle OAB is an equilateral triangle. Area of equilateral triangle = (3\sqrt{3}/4) × side² = (3\sqrt{3}/4) × 15² = (1.73/4) × 225 = 97.3125 cm²

  3. Step 3: Calculate Area of Minor Segment Area of minor segment = Area of sector – Area of triangle = 117.75 – 97.3125 = 20.4375 cm² ≈\approx 20.44 cm²

  4. Step 4: Calculate Area of Major Segment Area of circle = πr² = 3.14×2253.14 \times 225 = 706.5 cm² Area of major segment = Area of circle – Area of minor segment = 706.5 – 20.4375 = 686.0625 cm² ≈\approx 686.06 cm²

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