STEP 1: find side $ a $

To find side $ a $ use formula:

After substituting $B = 6\, \text{cm}$ we have:

$$ B = a^2 $$ $$ a^2 = 6\, \text{cm} $$ $$ a = \sqrt{ 6\, \text{cm} } $$

STEP 2: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $a = \sqrt{ 6 }\, \text{cm}^0$ we have:

$$ d = \sqrt{ 2 } \cdot \sqrt{ 6 } $$ $$ d = 2 \sqrt{ 3 } $$

STEP 3: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $d = 2 \sqrt{ 3 }\, \text{cm}^0$ and $e = 10\, \text{cm}$ we have:

$$ h ^ {\,2} + \frac{ 2 \sqrt{ 3 } }{ 4 } = \left( 10\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + \frac{ 12 }{ 4 } = \left( 10\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + 3 = \left( 10\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} = 100\, \text{cm}^2 - 3 $$ $$ h ^ {\,2} = 97\, \text{cm}^2 $$ $$ h = \sqrt{ 97\, \text{cm}^2 } $$$$ h = \sqrt{ 97 }\, \text{cm} $$

STEP 4: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $a = \sqrt{ 6 }\, \text{cm}^0$ and $h = \sqrt{ 97 }\, \text{cm}$ we have:

$$ V = \dfrac{ 6 \cdot \sqrt{ 97 }\, \text{cm} }{ 3 }$$$$ V = \dfrac{ 6 \sqrt{ 97 }\, \text{cm} }{ 3 } $$$$ V = 2 \sqrt{ 97 }\, \text{cm} $$

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