STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $d = 12\, \text{cm}$ we have:

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

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $d = 12\, \text{cm}$ and $e = 10\, \text{cm}$ we have:

$$ h ^ {\,2} + \frac{ \left( 12\, \text{cm} \right)^{2} }{ 4 } = \left( 10\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + \frac{ 144\, \text{cm}^2 }{ 4 } = \left( 10\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + 36\, \text{cm}^2 = \left( 10\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} = 100\, \text{cm}^2 - 36\, \text{cm}^2 $$ $$ h ^ {\,2} = 64\, \text{cm}^2 $$ $$ h = \sqrt{ 64\, \text{cm}^2 } $$$$ h = 8\, \text{cm} $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

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

$$ V = \dfrac{ 72\, \text{cm}^2 \cdot 8\, \text{cm} }{ 3 }$$$$ V = \dfrac{ 576\, \text{cm}^3 }{ 3 } $$$$ V = 192\, \text{cm}^3 $$

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