Question

ABCD is a square. With centres A, B, C and D, four quadrants (each touching two of the remaining three) are drawn inside the square ABCD as shown in the figure. If the area of the shaded region is 42 cm², find the side of the square ABCD. Also, find the perimeter of the shaded region.

Answer 1

Area of shaded region = Area of square – Area of 4 quadrants

`42 = ("side")^2 - 4((πr^2)/4)`

42 = (2r)² – πr²

42 – 4r² – πr²

42 = r²(4 – π)

`42 = r^2(4 - 22/7)`

`42 = r^2((28 - 22)/7)`   ...(LCM = 7)

`42 = r^2(6/7)`

`(42 xx 7)/6 = r^2`

7 × 7 = r²

r = 7

∴ Side of square = r + r

= 7 + 7

= 14 cm

Perimeter of shaded region = Circumference of circle

= 2πR

= `2 xx 22/7 xx 7`

= 44 cm