STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

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

After substituting $h = 30\, \text{cm}$ and $s = 32\, \text{cm}$ we have:

$$ \left( 30\, \text{cm} \right)^{2} + \frac{ a^2 }{ 4 } = \left( 32\, \text{cm} \right)^{2} $$ $$ \frac{ a^2 }{ 4 } = \left( 32\, \text{cm} \right)^{2} - \left( 30\, \text{cm} \right)^{2} $$ $$ \frac{ a^2 }{ 4 } = 1024\, \text{cm}^2 - 900\, \text{cm}^2 $$ $$ a^2 = 124\, \text{cm}^2 \cdot 4 $$ $$ a^2 = 496\, \text{cm}^2 $$ $$ a = \sqrt{ 496\, \text{cm}^2 } $$$$ a = 4 \sqrt{ 31 }\, \text{cm} $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 4 \sqrt{ 31 }\, \text{cm}$ we have:

$$ B = \left( 4 \sqrt{ 31 }\, \text{cm} \right)^{2} $$ $$ B = 496\, \text{cm}^2 $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $a = 4 \sqrt{ 31 }\, \text{cm}$ and $s = 32\, \text{cm}$ we have:

$$ L = 8 \sqrt{ 31 }\, \text{cm} \cdot 32\, \text{cm} $$$$ L = 256 \sqrt{ 31 }\, \text{cm}^2 $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $B = 496\, \text{cm}^2$ and $L = 256 \sqrt{ 31 }\, \text{cm}^2$ we have:

$$ A = 496\, \text{cm}^2 + 256 \sqrt{ 31 }\, \text{cm}^2 $$ $$ A = 496\, \text{cm}^2 + 256 \sqrt{ 31 }\, \text{cm}^2 $$ $$ A = 1921.3477\, \text{cm}^2 $$

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