Find the volume $ V $ of a pyramid if side $a = 9$ and slant height $s = 10$.

$V = \frac{ 27 \sqrt{ 319}}{ 2 }$

STEP 1: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ a = 9 $ and $ s = 10 $ we have:

$$ h ^ {\,2} + \frac{ 9^2 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + \frac{ 81 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + \frac{ 81 }{ 4 } = 10^2 $$ $$ h ^ {\,2} = 100 - \frac{ 81 }{ 4 } $$ $$ h ^ {\,2} = \frac{ 319 }{ 4 } $$ $$ h = \sqrt{ \frac{ 319 }{ 4 } } $$$$ h = \frac{\sqrt{ 319 }}{ 2 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 9 $ and $ h = \frac{\sqrt{ 319 }}{ 2 } $ we have:

$$ V = \dfrac{ 81 \cdot \frac{\sqrt{ 319 }}{ 2 } }{ 3 }$$$$ V = \dfrac{ \frac{ 81 \sqrt{ 319}}{ 2 } }{ 3 } $$$$ V = \frac{ 27 \sqrt{ 319}}{ 2 } $$

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