Find the area $ A $ of a pyramid if side $a = 2$ and volume $V = 8$.

$A = 28.3311$

STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 2 $ we have:

$$ B = 2^2 $$ $$ B = 4 $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ V = 8 $ and $ a = 2 $ we have:

$$ 8 = \dfrac{ 2 ^{ 2 } \cdot h }{ 3 } $$$$ 8 \cdot 3 = 2 ^{ 2 } \cdot h $$$$ 24 = 4 \cdot h $$$$ h = \dfrac{ 24 }{ 4 } $$$$ h = 6 $$

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 = 6 $ and $ a = 2 $ we have:

$$ 6^2 + \frac{ 2^2 }{ 4 }= s^2 $$ $$ 36 + \frac{ 4 }{ 4 }= s^2 $$ $$ 36 + 1 = s^2 $$ $$ s^2 = 37 $$ $$ s = \sqrt{ 37 } $$

STEP 4: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 2 $ and $ s = \sqrt{ 37 } $ we have:

$$ L = 4 \cdot \sqrt{ 37 } $$$$ L = 4 \sqrt{ 37 } $$

STEP 5: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 4 $ and $ L = 4 \sqrt{ 37 } $ we have:

$$ A = 4 + 4 \sqrt{ 37 } $$ $$ A = 4 + 4 \sqrt{ 37 } $$ $$ A = 28.3311 $$

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