Find the volume $ V $ of a pyramid if side $a = 12$ and slant height $s = 11$.

$V = 48 \sqrt{ 85 }$

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 = 11 $ we have:

$$ h ^ {\,2} + \frac{ 12^2 }{ 4 } = 11^2 $$ $$ h ^ {\,2} + \frac{ 144 }{ 4 } = 11^2 $$ $$ h ^ {\,2} + 36 = 11^2 $$ $$ h ^ {\,2} = 121 - 36 $$ $$ h ^ {\,2} = 85 $$ $$ h = \sqrt{ 85 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 12 $ and $ h = \sqrt{ 85 } $ we have:

$$ V = \dfrac{ 144 \cdot \sqrt{ 85 } }{ 3 }$$$$ V = \dfrac{ 144 \sqrt{ 85 } }{ 3 } $$$$ V = 48 \sqrt{ 85 } $$

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