>>256
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In[1]:= r=6

Out[1]= 6

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

Out[2]= 4 (9 - 9 Sqrt[3] + 3 Pi)

モンテカルロ法での検証
r=6
f[p_,o_] := Total[(p-o)^2] <= r^2
f4[p_] :=(
AllTrue[{f[p,{0,0}],f[p,{r,0}],f[p,{r,r}],f[p,{0,r}]},#==True&]
)
ratio=f4 /@ Tuples[Range[0,r,0.001],2] // Boole // Mean;
r^2 * ratio // N