Find Area of Area Not Filled In By Circle If I have a square with four quarters of a circle, how can I find the area not filled in by the circles? What I'm asking is here:

My drawing is har from perfect, but I think you get what I mean. I'm trying to find the area of the space where the arrow is pointing. Here are my thoughts on the problem:
The square is 24 inches by 24 inches, meaning its area is 576. Four quarters of a circle make a circle, and the radius of the circle is 12. To find the area of the circle, I do pi*12^2, which is 144pi. So does that mean the area of the space in between the semicircles is 576 - 144pi, or can the answer be further simplified without doing 144 * pi?
You are correct and there is no simplification for $576-144\pi$.  You can feed it to a calculator and get an approximate answer, which could be 
$$123.61065788306977366137935280775158467560760648998\\
4761779607978707674437494785904197562985150735142210570\\
6755420873478428184863762247161586315482452294987190871\\
590892109208529607994111742953504900249971222512195041055\\8375749950...$$
if you ask Alpha and click more digits a couple times.  It will give you more yet if you want.
Here u go,

This is made up of four 90 degrees sectors and a small part that u want to find the area.
So what u must do is...

||  AREA OF SQUARE - 4(AREA OF 90 DEGREES            SECTOR ||

So now lets find the value,

1.Finding the area of square :-

    24x24 = 576


2.Finding the area of 90 degrees sector :-

So as u mentioned the radius is 12 therefore,

\frac{90}{360}x\pixr^2 
   = 113.0973355

3.Substitution

AREA OF SQUARE - 4(AREA OF 90 DEGREES                       SECTOR  

576 - 4(113.0973355)

576 - 452.3893421= 123.6106579

Finally your answer is..
   123.6106579