$\begin{array}{1 1}r=R(0.414)\\r=R(1.414)\\r=R(2.414)\\r=R(0.214)\end{array} $

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A sphere fitting into a octahedral void is shown by shaded circle.ABC is a right angle triangle.Applying Phythagoras theorem,

$BC^2=AB^2+AC^2$

$(2R)^2=(R+r)^2+(R+r)^2$

$\Rightarrow 2(R+r)^2$

(or) $\large\frac{(2R)^2}{2}=$$(R+r)^2$

$\sqrt{\large\frac{(2R)^2}{2}}=$$R+r$

(or) $\sqrt{\large\frac{4R^2}{2}}$$=R+r$

$\sqrt{2R^2}=R+r$

$\sqrt 2R=R+r$

$r=\sqrt 2R-R$

$r=R(\sqrt 2-1)$

$\;\;=R(1.414-1)$

$r=R(0.414)$

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