Find the area $ A $ of a pyramid if side $a = 2$ and lateral edge $e = 12$.

$A = 51.833$

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 slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $ a = 2 $ and $ e = 12 $ we have:

$$ s ^ {\,2} + \frac{ 2^2 }{ 4 } = 12^2 $$ $$ s ^ {\,2} + \frac{ 4 }{ 4 } = 12^2 $$ $$ s ^ {\,2} + 1 = 12^2 $$ $$ s ^ {\,2} = 144 - 1 $$ $$ s ^ {\,2} = 143 $$ $$ s = \sqrt{ 143 } $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 2 $ and $ s = \sqrt{ 143 } $ we have:

$$ L = 4 \cdot \sqrt{ 143 } $$$$ L = 4 \sqrt{ 143 } $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 4 $ and $ L = 4 \sqrt{ 143 } $ we have:

$$ A = 4 + 4 \sqrt{ 143 } $$ $$ A = 4 + 4 \sqrt{ 143 } $$ $$ A = 51.833 $$

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