STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 2\, \text{cm}$ we have:

$$ B = \left( 2\, \text{cm} \right)^{2} $$ $$ B = 4\, \text{cm}^2 $$

STEP 2: find slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $a = 2\, \text{cm}$ and $e = 12\, \text{cm}$ we have:

$$ s ^ {\,2} + \frac{ \left( 2\, \text{cm} \right)^{2} }{ 4 } = \left( 12\, \text{cm} \right)^{2} $$ $$ s ^ {\,2} + \frac{ 4\, \text{cm}^2 }{ 4 } = \left( 12\, \text{cm} \right)^{2} $$ $$ s ^ {\,2} + 1\, \text{cm}^2 = \left( 12\, \text{cm} \right)^{2} $$ $$ s ^ {\,2} = 144\, \text{cm}^2 - 1\, \text{cm}^2 $$ $$ s ^ {\,2} = 143\, \text{cm}^2 $$ $$ s = \sqrt{ 143\, \text{cm}^2 } $$$$ s = \sqrt{ 143 }\, \text{cm} $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $a = 2\, \text{cm}$ and $s = \sqrt{ 143 }\, \text{cm}$ we have:

$$ L = 4\, \text{cm} \cdot \sqrt{ 143 }\, \text{cm} $$$$ L = 4 \sqrt{ 143 }\, \text{cm}^2 $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $B = 4\, \text{cm}^2$ and $L = 4 \sqrt{ 143 }\, \text{cm}^2$ we have:

$$ A = 4\, \text{cm}^2 + 4 \sqrt{ 143 }\, \text{cm}^2 $$ $$ A = 4\, \text{cm}^2 + 4 \sqrt{ 143 }\, \text{cm}^2 $$ $$ A = 51.833\, \text{cm}^2 $$

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