Find the volume $ V $ of a pyramid if lateral edge $e = 16$ and base diagonal $d = 16.28$.

$V = 608.4676$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d = 16.28 $ we have:

$$ 16.28 = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 16.28 }{ \sqrt{ 2 } } $$ $$ a \approx 11.5117 $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 16.28 $ and $ e = 16 $ we have:

$$ h ^ {\,2} + \frac{ 16.28^2 }{ 4 } = 16^2 $$ $$ h ^ {\,2} + \frac{ 265.0384 }{ 4 } = 16^2 $$ $$ h ^ {\,2} + 66.2596 = 16^2 $$ $$ h ^ {\,2} = 256 - 66.2596 $$ $$ h ^ {\,2} = 189.7404 $$ $$ h = \sqrt{ 189.7404 } $$$$ h = 13.7746 $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 11.5117 $ and $ h = 13.7746 $ we have:

$$ V = \dfrac{ 132.5192 \cdot 13.7746 }{ 3 }$$$$ V = \dfrac{ 1825.4028 }{ 3 } $$$$ V = 608.4676 $$

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