STEP 1: find radius $ r $

To find radius $ r $ use formula:

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

After substituting $V = 252\pi\, \text{cm}$ and $h = 21\, \text{cm}$ we have:

$$ 252\pi\, \text{cm} = \dfrac{ 21\, \text{cm} ^{ 2 } \cdot r \cdot \pi}{ 3 } $$$$ 252\pi\, \text{cm} \cdot 3 = 21\, \text{cm} ^{ 2 } \cdot r \cdot \pi $$$$ 756\pi\, \text{cm} = 441\, \text{cm}^2 \cdot r \cdot \pi $$$$ r = \dfrac{ 756\pi\, \text{cm} }{ 441\, \text{cm}^2 \, \pi} $$$$ r = \frac{ 12 }{ 7 }\, \text{cm}^-1 $$

STEP 2: find side $ l $

To find side $ l $ use Pythagorean Theorem:

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

After substituting $r = \dfrac{ 12 }{ 7 }\, \text{cm}^-1$ and $h = 21\, \text{cm}$ we have:

$$ \left( \frac{ 12 }{ 7 }\, \text{cm}^-1 \right)^{2} + \left( 21\, \text{cm} \right)^{2} = l^2 $$ $$ \frac{ 144 }{ 49 }\, \text{cm}^-2 + 441\, \text{cm}^2 = l^2 $$ $$ l^2 = \frac{ 21753 }{ 49 }\, \text{cm}^-2 $$ $$ l = \sqrt{ \frac{ 21753 }{ 49 }\, \text{cm}^-2 } $$$$ l = \frac{ 3 \sqrt{ 2417}}{ 7 }\, \text{cm}^-1 $$

STEP 3: find Curved Surface Area $ CSA $

To find Curved Surface Area $ CSA $ use formula:

$$ CSA = l \cdot r \cdot \pi$$

After substituting $r = \dfrac{ 12 }{ 7 }\, \text{cm}^-1$ and $l = \dfrac{ 3 \sqrt{ 2417}}{ 7 }\, \text{cm}^-1$ we have:

$$ CSA = \frac{ 3 \sqrt{ 2417}}{ 7 }\, \text{cm}^-1 \cdot \frac{ 12 }{ 7 }\, \text{cm}^-1 \cdot \pi$$$$ CSA = \frac{ 3 \sqrt{ 2417}}{ 7 }\, \text{cm}^-1 \cdot \frac{ 12 }{ 7 }\, \text{cm}^-1 \cdot \pi $$$$ CSA = \frac{ 36 \sqrt{ 2417}}{ 49 }\pi\, \text{cm}^-2 $$

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