STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $h = 8\, \text{cm}$ and $s = 12\, \text{cm}$ we have:

$$ \left( 8\, \text{cm} \right)^{2} + \frac{ a^2 }{ 4 } = \left( 12\, \text{cm} \right)^{2} $$ $$ \frac{ a^2 }{ 4 } = \left( 12\, \text{cm} \right)^{2} - \left( 8\, \text{cm} \right)^{2} $$ $$ \frac{ a^2 }{ 4 } = 144\, \text{cm}^2 - 64\, \text{cm}^2 $$ $$ a^2 = 80\, \text{cm}^2 \cdot 4 $$ $$ a^2 = 320\, \text{cm}^2 $$ $$ a = \sqrt{ 320\, \text{cm}^2 } $$$$ a = 8 \sqrt{ 5 }\, \text{cm} $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $a = 8 \sqrt{ 5 }\, \text{cm}$ and $h = 8\, \text{cm}$ we have:

$$ V = \dfrac{ 320\, \text{cm}^2 \cdot 8\, \text{cm} }{ 3 }$$$$ V = \dfrac{ 2560\, \text{cm}^3 }{ 3 } $$$$ V = \frac{ 2560 }{ 3 }\, \text{cm}^3 $$

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