Question

Calculate the area of the shaded portion. The quadrants shown in the figure are each of radius 7 cm.

Answer 1

Given:

• Four quadrants each of radius 7 cm.
• The shaded region is the overlapping central part formed by these quadrants.

Step-wise calculation:

1. Each quadrant has radius r = 7 cm.

2. The shape formed by the quadrants is symmetrical the points of intersection create a square inside the figure.

3. The diagonal of this square corresponds to the diameter of the quadrants, which is 14 cm.

4. Using the square diagonal formula:

Side of square = `"Diagonal"/sqrt(2)`

= `14/sqrt(2)`

= `7sqrt(2)` cm

5. Area of the square = Side²

= `(7sqrt(2))^2`

= 49 × 2

= 98 cm²

6. The shaded region is only part of the square, specifically the area within the quadrant arcs.

7. The shaded area can be calculated as:

Shaded area = Area of square – Area outside the quadrants

8. The part outside the intersection corresponds to four small circular segments.

9. Each segment’s area = area of quadrant sector – area of the right triangle formed by radius lines.

10. Area of one quadrant:

`1/4 πr^2 = 1/4 xx π xx 7^2`

= `(49π)/4`

11. Area of right triangle inside quadrant: 

`1/2 xx 7 xx 7 = 49/2`

12. Area of one segment:

Segment area = `(49π)/4 - 49/2`

= `49/4 (π - 2)`

13. Total area of four such segments: 

4 × Segment area

= `4 xx 49/4 (π - 2)` 

= 49(π – 2)

14. Since the shaded portion is the total area of the square minus these four segments:

Shaded area = 98 – 49(π – 2)

= 98 – 49π + 98

= 196 – 49π

15. Using π ≈ 3.14:

196 – 49 × 3.14

= 196 – 153.86

= 42.14 cm²

The area of the shaded portion is approximately 42 cm².