STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

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

After substituting $h = 6\, \text{cm}$ and $s = 12\, \text{cm}$ we have:

$$ \left( 6\, \text{cm} \right)^{2} + \frac{ a^2 }{ 4 } = \left( 12\, \text{cm} \right)^{2} $$ $$ \frac{ a^2 }{ 4 } = \left( 12\, \text{cm} \right)^{2} - \left( 6\, \text{cm} \right)^{2} $$ $$ \frac{ a^2 }{ 4 } = 144\, \text{cm}^2 - 36\, \text{cm}^2 $$ $$ a^2 = 108\, \text{cm}^2 \cdot 4 $$ $$ a^2 = 432\, \text{cm}^2 $$ $$ a = \sqrt{ 432\, \text{cm}^2 } $$$$ a = 12 \sqrt{ 3 }\, \text{cm} $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 12 \sqrt{ 3 }\, \text{cm}$ we have:

$$ B = \left( 12 \sqrt{ 3 }\, \text{cm} \right)^{2} $$ $$ B = 432\, \text{cm}^2 $$

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