Find the volume $ V $ of a cone if radius $r = 3\, \text{cm}$ and slant height $l = 6\, \text{cm}$.

$27\pi\, \text{cm}^3$

STEP 1: find side $ h $

To find side $ h $ use Pythagorean Theorem:

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

After substituting $r = 3\, \text{cm}$ and $l = 6\, \text{cm}$ we have:

$$ \left( 3\, \text{cm} \right)^{2} + h^2 = \left( 6\, \text{cm} \right)^{2} $$ $$ h^2 = \left( 6\, \text{cm} \right)^{2} - \left( 3\, \text{cm} \right)^{2} $$ $$ h^2 = 36\, \text{cm}^2 - 9\, \text{cm}^2 $$ $$ h^2 = 27\, \text{cm}^2 $$ $$ h = \sqrt{ 27\, \text{cm}^2 } $$$$ h = 3 \sqrt{ 3 }\, \text{cm} $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $r = 3\, \text{cm}$ and $h = 3 \sqrt{ 3 }\, \text{cm}$ we have:

$$ V = \dfrac{ 27\, \text{cm}^2 \cdot 3\, \text{cm} \cdot \pi}{ 3 }$$$$ V = \dfrac{ 81\, \text{cm}^3 \, \pi}{ 3 } $$$$ V = 27\pi\, \text{cm}^3 $$

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