STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 8\, \text{cm}$ we have:

$$ B = \left( 8\, \text{cm} \right)^{2} $$ $$ B = 64\, \text{cm}^2 $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $V = 64\, \text{cm}$ and $a = 8\, \text{cm}$ we have:

$$ 64\, \text{cm} = \dfrac{ 8\, \text{cm} ^{ 2 } \cdot h }{ 3 } $$$$ 64\, \text{cm} \cdot 3 = 8\, \text{cm} ^{ 2 } \cdot h $$$$ 192\, \text{cm} = 64\, \text{cm}^2 \cdot h $$$$ h = \dfrac{ 192\, \text{cm} }{ 64\, \text{cm}^2 } $$$$ h = 3\, \text{cm}^-1 $$

STEP 3: find slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $h = 3\, \text{cm}^-1$ and $a = 8\, \text{cm}$ we have:

$$ \left( 3\, \text{cm}^-1 \right)^{2} + \frac{ \left( 8\, \text{cm} \right)^{2} }{ 4 }= s^2 $$ $$ 9\, \text{cm}^-2 + \frac{ 64\, \text{cm}^2 }{ 4 }= s^2 $$ $$ 9\, \text{cm}^-2 + 16\, \text{cm}^2 = s^2 $$ $$ s^2 = 25\, \text{cm}^-2 $$ $$ s = \sqrt{ 25\, \text{cm}^-2 } $$$$ s = 5\, \text{cm}^-1 $$

STEP 4: find lateral surface $ L $

To find lateral surface $ L $ use formula:

$$ L = 2 \cdot a \cdot s $$

After substituting $a = 8\, \text{cm}$ and $s = 5\, \text{cm}^-1$ we have:

$$ L = 16\, \text{cm} \cdot 5\, \text{cm}^-1 $$$$ L = 80 $$

STEP 5: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $B = 64\, \text{cm}^2$ and $L = 80\, \text{cm}^0$ we have:

$$ A = 64\, \text{cm}^2 + 80 $$ $$ A = 64\, \text{cm}^2 + 80 $$ $$ A = 144\, \text{cm}^2 $$

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