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 slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $a = 6 \sqrt{ 2 }\, \text{cm}$ and $e = 10\, \text{cm}$ we have:

$$ s ^ {\,2} + \frac{ \left( 6 \sqrt{ 2 }\, \text{cm} \right)^{2} }{ 4 } = \left( 10\, \text{cm} \right)^{2} $$ $$ s ^ {\,2} + \frac{ 72\, \text{cm}^2 }{ 4 } = \left( 10\, \text{cm} \right)^{2} $$ $$ s ^ {\,2} + 18\, \text{cm}^2 = \left( 10\, \text{cm} \right)^{2} $$ $$ s ^ {\,2} = 100\, \text{cm}^2 - 18\, \text{cm}^2 $$ $$ s ^ {\,2} = 82\, \text{cm}^2 $$ $$ s = \sqrt{ 82\, \text{cm}^2 } $$$$ s = \sqrt{ 82 }\, \text{cm} $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $a = 6 \sqrt{ 2 }\, \text{cm}$ and $s = \sqrt{ 82 }\, \text{cm}$ we have:

$$ L = 12 \sqrt{ 2 }\, \text{cm} \cdot \sqrt{ 82 }\, \text{cm} $$$$ L = 24 \sqrt{ 41 }\, \text{cm}^2 $$

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