Find the volume $ V $ of a pyramid if side $a = 12$ and slant height $s = 10$.

$V = 384$

STEP 1: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ a = 12 $ and $ s = 10 $ we have:

$$ h ^ {\,2} + \frac{ 12^2 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + \frac{ 144 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + 36 = 10^2 $$ $$ h ^ {\,2} = 100 - 36 $$ $$ h ^ {\,2} = 64 $$ $$ h = \sqrt{ 64 } $$$$ h = 8 $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 12 $ and $ h = 8 $ we have:

$$ V = \dfrac{ 144 \cdot 8 }{ 3 }$$$$ V = \dfrac{ 1152 }{ 3 } $$$$ V = 384 $$

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