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:

Four quarters of circle inside square

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?

  • 1
    $\begingroup$ 576-144pi is the correct area $\endgroup$ – Birch Bryant Jun 1 '17 at 5:00
  • $\begingroup$ @user253929 my question is if it can be further simplified without multiplying 144 and pi $\endgroup$ – Nathan Chan Jun 1 '17 at 5:01
  • $\begingroup$ A very different but related question: math.stackexchange.com/questions/1879509/… Well, maybe only because it also involves four circles of the same size. $\endgroup$ – Robert Soupe Jun 3 '17 at 14:39

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...


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



576 - 4(113.0973355)

576 - 452.3893421= 123.6106579

Finally your answer is.. 123.6106579

  • $\begingroup$ Welcome to MSE! Plese use MathJax to write math and please use proper enlgish in your posts (write "you" and not "u"). $\endgroup$ – yakobyd Jun 29 '18 at 3:18

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