Let's first find the outer shaded region, not the one in the square.
Square is 1cm
Two quarter circles equal pi/2 cm
The big quarter circle is pi cm
So the outer area is pi/2-1 cm
The inner part in the square is a 90 degree sector of a circle with radius 1 minus an isosceles right triangle with side 1 (two of them in total).
pi/4-1/2 times 2 equals pi/2-1
Add them up to get
\(\pi -2\) cm^2.
You are very welcome!
The area of (1/2) the shaded region inside the square is =
Area of a quarter circle with a radius of 1cm - area of a right triangle with legs of 1 cm =
pi (1)^2/ 4 - (1/2)(1)^2 =
( 1/2) [ pi/2 - 1]
So...usihg symmetry this shaded area = [ pi/2 - 1]
And the outer shaded area =
Area of a quarter circle with a radius of 2 cm - area of two quarter circes with a radius of 1 cm - square with a side of 1 cm =
pi * (2)^2 / 4 - 2* pi (1)^2 / 4 - 1 =
pi - (1/2)pi - 1 =
So....the toal shaded area =
2 [ pi/2 - 1 ] =
(pi - 2) cm^2