The relations of Young's modulus, $Y$, bulk modulus, $K$, the modulus of rigidity, $\eta$ and Poisson's ratio, $\sigma$ are given as

$Y=3K(1-2\sigma ) \dots \left(1\right)$ and 

$Y=2\eta \left(1+\sigma \right) \dots \left(2\right)$

From equation $\left(1\right) \mathrm{and} \left(2\right)$, we have

$\Rightarrow 2\eta \left(1+\sigma \right)=3K\left(1-2\sigma \right)$

$\Rightarrow \sigma =\left(\frac{3K-2\eta }{6K+2\eta }\right)$