STEP 1: find side $ a $

To find side $ a $ use formula:

$$ V = \dfrac{ a ^{ 2 } \cdot h }{ 3 } $$

After substituting $V = 182\, \text{cm}$ and $h = 13\, \text{cm}$ we have:

$$ 182\, \text{cm} = \dfrac{ a ^{ 2 } \cdot \left( 13\, \text{cm} \right)^{4} }{ 3 } $$$$ 182\, \text{cm} \cdot 3 = a ^{ 2 } \cdot \left( 13\, \text{cm} \right)^{4} $$$$ 546\, \text{cm} = 13\, \text{cm} \cdot a ^{ 2 } $$$$ a ^{ 2 } = \dfrac{ 546\, \text{cm}}{ 13\, \text{cm} } $$$$ a ^{ 2 } \approx 13.369 $$$$ a \approx \sqrt{ 13.369 } $$$$ a \approx 3.6564 $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 3.6564\, \text{cm}^0$ we have:

$$ B = 3.6564 $$ $$ B = 13.369 $$

STEP 3: find slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $h = 13\, \text{cm}$ and $a = 3.6564\, \text{cm}^0$ we have:

$$ \left( 13\, \text{cm} \right)^{2} + \frac{ 3.6564 }{ 4 }= s^2 $$ $$ 169\, \text{cm}^2 + \frac{ 13.369 }{ 4 }= s^2 $$ $$ 169\, \text{cm}^2 + 3.3422 = s^2 $$ $$ s^2 = 172.3422\, \text{cm}^2 $$ $$ s = \sqrt{ 172.3422\, \text{cm}^2 } $$$$ s = 13.1279\, \text{cm} $$

STEP 4: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $a = 3.6564\, \text{cm}^0$ and $s = 13.1279\, \text{cm}$ we have:

$$ L = 7.3127 \cdot 13.1279\, \text{cm} $$$$ L = 96.0008\, \text{cm} $$

STEP 5: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $B = 13.369\, \text{cm}^0$ and $L = 96.0008\, \text{cm}$ we have:

$$ A = 13.369 + 96.0008\, \text{cm} $$ $$ A = 13.369 + 96.0008\, \text{cm} $$ $$ A = 109.3698 $$

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