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

$B = 216$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use Pythagorean Theorem:

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

After substituting $ h = 6 $ and $ e = 12 $ we have:

$$ 6^2 + \frac{ d^2 }{ 4 } = 12^2 $$ $$ \frac{ d^2 }{ 4 } = 12^2 - 6^2 $$ $$ \frac{ d^2 }{ 4 } = 144 - 36 $$ $$ d^2 = 108 \cdot 4 $$ $$ d^2 = 432 $$ $$ d = \sqrt{ 432 } $$$$ d = 12 \sqrt{ 3 } $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d = 12 \sqrt{ 3 } $ we have:

$$ 12 \sqrt{ 3 } = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 12 \sqrt{ 3 } }{ \sqrt{ 2 } } $$ $$ a = 6 \sqrt{ 6 } $$

STEP 3: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

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

$$ B = \left(6 \sqrt{ 6 }\right)^2 $$ $$ B = 216 $$

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