Find the volume $ V $ of a cone if diameter $d = 6$ and slant height $l = 8$.

$V = 3 \sqrt{ 55 }\pi$

STEP 1: find radius $ r $

To find radius $ r $ use formula:

$$ d = 2 \cdot r $$

After substituting $ d = 6 $ we have:

$$ 6 = 2 \cdot r $$ $$ r = \dfrac{ 6 }{ 2 } $$ $$ r = 3 $$

STEP 2: find side $ h $

To find side $ h $ use Pythagorean Theorem:

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

After substituting $ r = 3 $ and $ l = 8 $ we have:

$$ 3^2 + h^2 = 8^2 $$ $$ h^2 = 8^2 - 3^2 $$ $$ h^2 = 64 - 9 $$ $$ h^2 = 55 $$ $$ h = \sqrt{ 55 } $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ r = 3 $ and $ h = \sqrt{ 55 } $ we have:

$$ V = \dfrac{ 9 \cdot \sqrt{ 55 } \cdot \pi}{ 3 }$$$$ V = \dfrac{ 9 \sqrt{ 55 } \, \pi}{ 3 } $$$$ V = 3 \sqrt{ 55 }\pi $$

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