STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 19.9\, \text{cm}$ we have:

$$ B = \left( 19.9\, \text{cm} \right)^{2} $$ $$ B = 396.01\, \text{cm}^2 $$

STEP 2: find slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $h = 8.2\, \text{cm}$ and $a = 19.9\, \text{cm}$ we have:

$$ \left( 8.2\, \text{cm} \right)^{2} + \frac{ \left( 19.9\, \text{cm} \right)^{2} }{ 4 }= s^2 $$ $$ 67.24\, \text{cm}^2 + \frac{ 396.01\, \text{cm}^2 }{ 4 }= s^2 $$ $$ 67.24\, \text{cm}^2 + 99.0025\, \text{cm}^2 = s^2 $$ $$ s^2 = 166.2425\, \text{cm}^2 $$ $$ s = \sqrt{ 166.2425\, \text{cm}^2 } $$$$ s = 12.8935\, \text{cm} $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $a = 19.9\, \text{cm}$ and $s = 12.8935\, \text{cm}$ we have:

$$ L = 39.8\, \text{cm} \cdot 12.8935\, \text{cm} $$$$ L = 513.1615\, \text{cm}^2 $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $B = 396.01\, \text{cm}^2$ and $L = 513.1615\, \text{cm}^2$ we have:

$$ A = 396.01\, \text{cm}^2 + 513.1615\, \text{cm}^2 $$ $$ A = 396.01\, \text{cm}^2 + 513.1615\, \text{cm}^2 $$ $$ A = 909.1715\, \text{cm}^2 $$

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