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In[1]:= 4 * Integrate[Sqrt[r^2-x^2]-r/2,{x,r/2,r*Sqrt[3]/2}]

2
r (-3 (-1 + Sqrt[3]) r + Pi Sqrt[r ])
Out[1]= -------------------------------------
3

In[2]:= r=6

Out[2]= 6

In[3]:= f[p_,o_] := Total[(p-o)^2] <= r^2

In[4]:= f4[p_] :=(
AllTrue[{f[p,{0,0}],f[p,{r,0}],f[p,{r,r}],f[p,{0,r}]},#==True&]
)

In[5]:= ratio=f4 /@ Tuples[Range[0,r,0.001],2] // Boole // Mean;

In[6]:= r^2 * ratio // N

Out[6]= 11.3415

In[7]:=
11.3415