In the square below the four quarter circles are congruent. find the area of the shaded region.

The quarter circle pieces are all congruent allowing us to glue them together to form a circle. The circle will be completely inside the square. This circle is as large as possible so it's not spilling outside of the square. 

The area of the square is 8*8 = 64 square units

The circle has a radius of 8/2 = 4 units. Use r = 4 and plug this into the area of a circle formula
A = pi*r^2
A = pi*4^2
A = pi*16
A = 16pi

Now subtract the two areas
(area of square) - (area of circle) = 64 - 16pi

The exact answer, in terms of pi, is 64 - 16pi

If your teacher wants you to find the approximate decimal version, then you would use a calculator to find that 64 - 16pi = 13.7345175 approximately. If you end up going with the approximate answer, be sure to follow the instructions on what your teacher wants with rounding.