Find the volume $ V $ of a pyramid if side $a = 8$ and lateral edge $e = 8$.

$V = \frac{ 256 \sqrt{ 2}}{ 3 }$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $ a = 8 $ we have:

$$ d = \sqrt{ 2 } \cdot 8 $$ $$ d = 8 \sqrt{ 2 } $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 8 \sqrt{ 2 } $ and $ e = 8 $ we have:

$$ h ^ {\,2} + \frac{ \left(8 \sqrt{ 2 }\right)^2 }{ 4 } = 8^2 $$ $$ h ^ {\,2} + \frac{ 128 }{ 4 } = 8^2 $$ $$ h ^ {\,2} + 32 = 8^2 $$ $$ h ^ {\,2} = 64 - 32 $$ $$ h ^ {\,2} = 32 $$ $$ h = \sqrt{ 32 } $$$$ h = 4 \sqrt{ 2 } $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 8 $ and $ h = 4 \sqrt{ 2 } $ we have:

$$ V = \dfrac{ 64 \cdot 4 \sqrt{ 2 } }{ 3 }$$$$ V = \dfrac{ 256 \sqrt{ 2 } }{ 3 } $$$$ V = \frac{ 256 \sqrt{ 2}}{ 3 } $$

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