Find the volume $ V $ of a pyramid if lateral edge $e = 16$ and base diagonal $d = 8.26$.

$V = 175.7746$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d = 8.26 $ we have:

$$ 8.26 = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 8.26 }{ \sqrt{ 2 } } $$ $$ a \approx 5.8407 $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 8.26 $ and $ e = 16 $ we have:

$$ h ^ {\,2} + \frac{ 8.26^2 }{ 4 } = 16^2 $$ $$ h ^ {\,2} + \frac{ 68.2276 }{ 4 } = 16^2 $$ $$ h ^ {\,2} + 17.0569 = 16^2 $$ $$ h ^ {\,2} = 256 - 17.0569 $$ $$ h ^ {\,2} = 238.9431 $$ $$ h = \sqrt{ 238.9431 } $$$$ h = 15.4578 $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 5.8407 $ and $ h = 15.4578 $ we have:

$$ V = \dfrac{ 34.1138 \cdot 15.4578 }{ 3 }$$$$ V = \dfrac{ 527.3238 }{ 3 } $$$$ V = 175.7746 $$

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