STEP 1: find height $ h $

To find height $ h $ use formula:

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

After substituting $V = 542.87\, \text{cm}$ and $r = 6\, \text{cm}$ we have:

$$ 542.87\, \text{cm} = \dfrac{ h ^{ 2 } \cdot \left( 6\, \text{cm} \right)^{4} \cdot \pi}{ 3 } $$$$ 542.87\, \text{cm} \cdot 3 = h ^{ 2 } \cdot \left( 6\, \text{cm} \right)^{4} \cdot \pi $$$$ 1628.61\, \text{cm} = 6\, \text{cm} \cdot h ^{ 2 } \cdot \pi $$$$ h ^{ 2 } = \dfrac{ 1628.61\, \text{cm}}{ 6\, \text{cm} \, \pi } $$$$ h ^{ 2 } \approx 86.4002 $$$$ h \approx \sqrt{ 86.4002 } $$$$ h \approx 9.2952 $$

STEP 2: find side $ l $

To find side $ l $ use Pythagorean Theorem:

$$ r^2 + h^2 = l^2 $$

After substituting $r = 6\, \text{cm}$ and $h = 9.2952\, \text{cm}^0$ we have:

$$ \left( 6\, \text{cm} \right)^{2} + 9.2952 = l^2 $$ $$ 36\, \text{cm}^2 + 86.4002 = l^2 $$ $$ l^2 = 122.4002\, \text{cm}^2 $$ $$ l = \sqrt{ 122.4002\, \text{cm}^2 } $$$$ l = 11.0635\, \text{cm} $$

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