To find height $ h $ use formula:

$$ V = \dfrac{ h ^{ 2 } \cdot r \cdot \pi}{ 3 } $$

After substituting $V = \dfrac{ 1 }{ 18 }\, \text{cm}$ and $r = \dfrac{ 1 }{ 3 }\, \text{cm}$ we have:

$$ \frac{ 1 }{ 18 }\, \text{cm} = \dfrac{ h ^{ 2 } \cdot \left( \frac{ 1 }{ 3 }\, \text{cm} \right)^{4} \cdot \pi}{ 3 } $$$$ \frac{ 1 }{ 18 }\, \text{cm} \cdot 3 = h ^{ 2 } \cdot \left( \frac{ 1 }{ 3 }\, \text{cm} \right)^{4} \cdot \pi $$$$ \frac{ 1 }{ 6 }\, \text{cm} = \frac{ 1 }{ 3 }\, \text{cm} \cdot h ^{ 2 } \cdot \pi $$$$ h ^{ 2 } = \dfrac{ \frac{ 1 }{ 6 }\, \text{cm}}{ \frac{ 1 }{ 3 }\, \text{cm} \, \pi } $$$$ h ^{ 2 } \approx 0.1592 $$$$ h \approx \sqrt{ 0.1592 } $$$$ h \approx 0.3989 $$

Cone Calculator