Find the volume $ V $ of a pyramid if lateral edge $e = 10$ and base diagonal $d = 12$.

$V = 192$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d = 12 $ we have:

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

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 12 $ and $ e = 10 $ we have:

$$ h ^ {\,2} + \frac{ 12^2 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + \frac{ 144 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + 36 = 10^2 $$ $$ h ^ {\,2} = 100 - 36 $$ $$ h ^ {\,2} = 64 $$ $$ h = \sqrt{ 64 } $$$$ h = 8 $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 6 \sqrt{ 2 } $ and $ h = 8 $ we have:

$$ V = \dfrac{ 72 \cdot 8 }{ 3 }$$$$ V = \dfrac{ 576 }{ 3 } $$$$ V = 192 $$

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