Find the Curved Surface Area $ CSA $ of a cone if height $h = 21$ and volume $V = 252\pi$.

$CSA = 18 \sqrt{ 53 }\pi$

STEP 1: find radius $ r $

To find radius $ r $ use formula:

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

After substituting $ V = 252\pi $ and $ h = 21 $ we have:

$$ 252\pi = \dfrac{ r ^{ 2 } \cdot 21 \cdot \pi}{ 3 } $$$$ 252\pi \cdot 3 = r ^{ 2 } \cdot 21 \cdot \pi $$$$ 756\pi = 21 \cdot r ^{ 2 } \cdot \pi $$$$ r ^{ 2 } = \dfrac{ 756\pi}{ 21 \, \pi } $$$$ r ^{ 2 } = 36 $$$$ r = \sqrt{ 36 } $$$$ r = 6 $$

STEP 2: find side $ l $

To find side $ l $ use Pythagorean Theorem:

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

After substituting $ r = 6 $ and $ h = 21 $ we have:

$$ 6^2 + 21^2 = l^2 $$ $$ 36 + 441 = l^2 $$ $$ l^2 = 477 $$ $$ l = \sqrt{ 477 } $$$$ l = 3 \sqrt{ 53 } $$

STEP 3: find Curved Surface Area $ CSA $

To find Curved Surface Area $ CSA $ use formula:

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

After substituting $ r = 6 $ and $ l = 3 \sqrt{ 53 } $ we have:

$$ CSA = 6 \cdot 3 \sqrt{ 53 } \cdot \pi$$$$ CSA = 6 \cdot 3 \sqrt{ 53 } \cdot \pi $$$$ CSA = 18 \sqrt{ 53 }\pi $$

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