Question

A square is inscribed in a circle. If the area of the shaded region is 224 cm², calculate

1. the radius 
2. area of the square.

Answer 1

Given: A square is inscribed in a circle and the area of the shaded region area of the circle minus area of the square is 224 cm².

Step-wise calculation:

1. Let:

Radius of the circle = r cm

Side of the square = s cm

Diagonal of the square = d cm

2. Since the square is inscribed in the circle, the diagonal of the square equals the diameter of the circle d = 2r

3. The area of the circle is Area of circle = πr²

4. The diagonal (d) relates to the side (s) of the square as:

`d = ssqrt(2)`

⇒ `s = d/sqrt(2)`

= `(2r)/sqrt(2)`

= `rsqrt(2)`

5. The area of the square is: 

Area of square = s²

= `(rsqrt(2))^2`

= 2r²

6. The shaded area, which is the area of the circle minus the area of the square, is given as πr² – 2r² = 224

7. Factor out (r²): r²(π – 2) = 224

8. Use `π = 22/7` for calculation:

`π - 2 = 22/7 - 2`

= `22/7 - 14/7`

= `8/7`

9. So, 

`r^2 xx 8/7 = 224`

⇒ `r^2 = (224 xx 7)/8`

= `224 xx 7/8`

= 224 × 0.875

= 196

10. Taking square root: r = `sqrt(196)` = 14 cm

11. Now, calculate the area of the square:

 Area of square = 2r²

= 2 × 196

= 392 cm²

Radius of the circle, r = 14 cm

Area of the square = 392 cm²