A few circles and a lone square!

Let the square be of a units.

Let the radius of the semi-circle be x units.

Let the radius of the inner circle be r units.

As it is given that the centre of the semicircles is the midpoint of the square's side, we can ascertain that a straight line (parallel to the side of a square) joining the midpoints will pass through the centre of the inner circle, thus formed.

So, we can form a connected equation: 2x + 2r = a [ Radii of two semicircles plus the diameter of the inner circle = Square's side]

Or, x + r = a/2 ... ( 1 )

Based on the perimeter condition given:

4/5 (Square's Perimeter) = Sum of the diameters of all the semi-circles

=> (4/5)(4a) = 4(2x)

=> a = 5x/2 ... ( 2 )

Using (1) and (2), we can get the ratio of r to x;

$\frac{r}{x}$ = $\frac{1}{4}$

We want to find the ratio of the area of inner circle (X) to that of the overall area of these four semicircles (Y);

X/Y = [ $\pi \cdot r^{2}$ ] ÷ [(4\2) $\pi \cdot x^{2}$ ]

= $1/2 \cdot [r^{2} \div x^{2}]$

Putting r/x = 1/4, we get;

= 1/32

Without losing generality, we can assume that one side of a square is of length 10 units. Then, the diameter of each one of the semi circles must be 8 units. Hence, their radius is 4 units.

Consider any pair of semi circles on opposite sides of the square, say which are along the vertical edges. Consider their common horizontal radial line which is also the diameter for the smaller circle at the square's center. The portion of this radial line which passes through the smaller circle is 10 - 4 - 4 = 2 units, which must be its diameter. Hence, the radius of the smaller circle is 1 unit.

Area of smaller circle : Total Area of 4 semi circles = $1^2$ : $4 * 4^2 / 2$ = 1 : 32