Find the volume $ V $ of a pyramid if side $a = 6$ and slant height $s = 9$.

$V = 72 \sqrt{ 2 }$

STEP 1: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ a = 6 $ and $ s = 9 $ we have:

$$ h ^ {\,2} + \frac{ 6^2 }{ 4 } = 9^2 $$ $$ h ^ {\,2} + \frac{ 36 }{ 4 } = 9^2 $$ $$ h ^ {\,2} + 9 = 9^2 $$ $$ h ^ {\,2} = 81 - 9 $$ $$ h ^ {\,2} = 72 $$ $$ h = \sqrt{ 72 } $$$$ h = 6 \sqrt{ 2 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 6 $ and $ h = 6 \sqrt{ 2 } $ we have:

$$ V = \dfrac{ 36 \cdot 6 \sqrt{ 2 } }{ 3 }$$$$ V = \dfrac{ 216 \sqrt{ 2 } }{ 3 } $$$$ V = 72 \sqrt{ 2 } $$

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