Find the height $ h $ of a pyramid if side $a = 6$ and lateral edge $e = 12$.

$h = 3 \sqrt{ 14 }$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

$$ d = \sqrt{ 2 } \cdot a $$

After substituting $ a = 6 $ we have:

$$ d = \sqrt{ 2 } \cdot 6 $$ $$ d = 6 \sqrt{ 2 } $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

$$ h^2 + \frac{ d^2 }{ 4 } = e^2 $$

After substituting $ d = 6 \sqrt{ 2 } $ and $ e = 12 $ we have:

$$ h ^ {\,2} + \frac{ \left(6 \sqrt{ 2 }\right)^2 }{ 4 } = 12^2 $$ $$ h ^ {\,2} + \frac{ 72 }{ 4 } = 12^2 $$ $$ h ^ {\,2} + 18 = 12^2 $$ $$ h ^ {\,2} = 144 - 18 $$ $$ h ^ {\,2} = 126 $$ $$ h = \sqrt{ 126 } $$$$ h = 3 \sqrt{ 14 } $$

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