Find the volume $ V $ of a pyramid if side $a = 4$ and slant height $s = 13$.

$V = \frac{ 16 \sqrt{ 165}}{ 3 }$

STEP 1: find height $ h $

To find height $ h $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $ a = 4 $ and $ s = 13 $ we have:

$$ h ^ {\,2} + \frac{ 4^2 }{ 4 } = 13^2 $$ $$ h ^ {\,2} + \frac{ 16 }{ 4 } = 13^2 $$ $$ h ^ {\,2} + 4 = 13^2 $$ $$ h ^ {\,2} = 169 - 4 $$ $$ h ^ {\,2} = 165 $$ $$ h = \sqrt{ 165 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 4 $ and $ h = \sqrt{ 165 } $ we have:

$$ V = \dfrac{ 16 \cdot \sqrt{ 165 } }{ 3 }$$$$ V = \dfrac{ 16 \sqrt{ 165 } }{ 3 } $$$$ V = \frac{ 16 \sqrt{ 165}}{ 3 } $$

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