Find the area $ A $ of a pyramid if side $a = 8$ and volume $V = 64$.

$A = 144$

STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 8 $ we have:

$$ B = 8^2 $$ $$ B = 64 $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ V = 64 $ and $ a = 8 $ we have:

$$ 64 = \dfrac{ 8 ^{ 2 } \cdot h }{ 3 } $$$$ 64 \cdot 3 = 8 ^{ 2 } \cdot h $$$$ 192 = 64 \cdot h $$$$ h = \dfrac{ 192 }{ 64 } $$$$ h = 3 $$

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 = 3 $ and $ a = 8 $ we have:

$$ 3^2 + \frac{ 8^2 }{ 4 }= s^2 $$ $$ 9 + \frac{ 64 }{ 4 }= s^2 $$ $$ 9 + 16 = s^2 $$ $$ s^2 = 25 $$ $$ s = \sqrt{ 25 } $$$$ s = 5 $$

STEP 4: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 8 $ and $ s = 5 $ we have:

$$ L = 16 \cdot 5 $$$$ L = 80 $$

STEP 5: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 64 $ and $ L = 80 $ we have:

$$ A = 64 + 80 $$ $$ A = 64 + 80 $$ $$ A = 144 $$

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