Find the area $ A $ of a pyramid if height $h = 13$ and volume $V = 182$.

$A = 215.6548$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ V = 182 $ and $ h = 13 $ we have:

$$ 182 = \dfrac{ a ^{ 2 } \cdot 13 }{ 3 } $$$$ 182 \cdot 3 = a ^{ 2 } \cdot 13 $$$$ 546 = 13 \cdot a ^{ 2 } $$$$ a ^{ 2 } = \dfrac{ 546}{ 13 } $$$$ a ^{ 2 } = 42 $$$$ a = \sqrt{ 42 } $$$$ a = \sqrt{ 42 } $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = \sqrt{ 42 } $ we have:

$$ B = \left( \sqrt{ 42 } \right)^{ 2 } $$ $$ B = 42 $$

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 $ and $ a = \sqrt{ 42 } $ we have:

$$ 13^2 + \frac{ \left( \sqrt{ 42 } \right)^{ 2 } }{ 4 }= s^2 $$ $$ 169 + \frac{ 42 }{ 4 }= s^2 $$ $$ 169 + \frac{ 21 }{ 2 } = s^2 $$ $$ s^2 = \frac{ 359 }{ 2 } $$ $$ s = \sqrt{ \frac{ 359 }{ 2 } } $$$$ s = \frac{\sqrt{ 718 }}{ 2 } $$

STEP 4: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = \sqrt{ 42 } $ and $ s = \frac{\sqrt{ 718 }}{ 2 } $ we have:

$$ L = 2 \sqrt{ 42 } \cdot \frac{\sqrt{ 718 }}{ 2 } $$$$ L = 2 \sqrt{ 7539 } $$

STEP 5: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 42 $ and $ L = 2 \sqrt{ 7539 } $ we have:

$$ A = 42 + 2 \sqrt{ 7539 } $$ $$ A = 42 + 2 \sqrt{ 7539 } $$ $$ A = 215.6548 $$

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