Answer : (a) = \(\frac{2+π(\sqrt{2}-2)}{(π-2)}\)
Area of square ABCD = Area of square AEFG = 16 cm2
Area of quadrant ADMB = \(\frac{1}{4}\) × π × (4)2 = 4π cm2
Radius of the smaller quadrant CPMQ = CM = AC – AM
=4√2 - 4 = 4 ( √2 - 1) cm.
∴ Area of the smaller quadrant
Area of the shaded region inside the square ABCD
= Area of sq. ABCD – (Area of quadrant ADMB + Area of quadrant CPMQ)
= 16 – [4π + 4π(3 – 2 √2 )]
= 16 – [4π (1 + 3 – 2 √2 )]
= 16 – [4π (4 – 2 √2 )]
= 8 [2 – 2π + √2 π]
Now area of quadrants AEG and EFG
= 2 × area of quadrant AEG = 2 × \(\frac{1}{4}\) × π × (4)2 = 8π
∴ Area of shaded region inside the square AEFG = Sum of the areas of quadrants AEG and EFG – Area of square AEFG
= 8π – 16 = 8(π – 2)
∴ Required ratio = \(\frac{8(2- 2π + \sqrt{2}\pi)}{8(π-2)}\)
= \(\frac{2+π(\sqrt{2}-2)}{(π-2)}\)