Given Data
The radius (r) of the circle = 20 cm
Angle subtended at the center (θ) = 90 degrees
Step 1: Calculate the Area of the Circle
The formula for the area of a circle is:
Area = πr²
- Area = π × (20 cm)²
- Area = π × 400 cm²
- Area ≈ 1256.64 cm² (using π ≈ 3.14)
Step 2: Calculate the Area of the Sector
The area of a sector is given by:
Area of Sector = (θ/360) × Area of Circle
- Area of Sector = (90/360) × 1256.64 cm²
- Area of Sector = (1/4) × 1256.64 cm²
- Area of Sector ≈ 314.16 cm²
Step 3: Calculate the Area of the Triangle
The triangle formed by the radius and the chord can be calculated using the formula:
Area = (1/2) × base × height
- The base (chord) can be found using the formula:
Chord = 2 × r × sin(θ/2)
- Chord = 2 × 20 cm × sin(45°)
- Chord = 2 × 20 cm × (√2/2)
- Chord = 20√2 cm
Now, the height of the triangle can be found as:
Height = r × cos(θ/2) = 20 cm × cos(45°) = 20 cm × (√2/2) = 10√2 cm
- Therefore, Area of Triangle = (1/2) × (20√2 cm) × (10√2 cm)
- Area of Triangle = 100 cm²
Step 4: Calculate the Area of the Major Segment
The area of the major segment can be calculated as:
Area of Major Segment = Area of Circle - (Area of Sector + Area of Triangle)
- Area of Major Segment = 1256.64 cm² - (314.16 cm² + 100 cm²)
- Area of Major Segment = 1256.64 cm² - 414.16 cm²
- Area of Major Segment ≈ 842.48 cm²
Final Answer
The area of the corresponding major segment of the circle is approximately 842.48 cm².