To find height $ h $ use formula:

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

After substituting $V = 27\pi\, \text{cm}$ and $r = 3\, \text{cm}$ we have:

$$ 27\pi\, \text{cm} = \dfrac{ h ^{ 2 } \cdot \left( 3\, \text{cm} \right)^{4} \cdot \pi}{ 3 } $$$$ 27\pi\, \text{cm} \cdot 3 = h ^{ 2 } \cdot \left( 3\, \text{cm} \right)^{4} \cdot \pi $$$$ 81\pi\, \text{cm} = 3\, \text{cm} \cdot h ^{ 2 } \cdot \pi $$$$ h ^{ 2 } = \dfrac{ 81\pi\, \text{cm}}{ 3\, \text{cm} \, \pi } $$$$ h ^{ 2 } = 27 $$$$ h = \sqrt{ 27 } $$$$ h = 3 \sqrt{ 3 } $$

Cone Calculator