Find the area $ A $ of a pyramid if side $a = 116$ and lateral edge $e = 86$.

$A = 28187.5433$

STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 116 $ we have:

$$ B = 116^2 $$ $$ B = 13456 $$

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 = 116 $ and $ e = 86 $ we have:

$$ s ^ {\,2} + \frac{ 116^2 }{ 4 } = 86^2 $$ $$ s ^ {\,2} + \frac{ 13456 }{ 4 } = 86^2 $$ $$ s ^ {\,2} + 3364 = 86^2 $$ $$ s ^ {\,2} = 7396 - 3364 $$ $$ s ^ {\,2} = 4032 $$ $$ s = \sqrt{ 4032 } $$$$ s = 24 \sqrt{ 7 } $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 116 $ and $ s = 24 \sqrt{ 7 } $ we have:

$$ L = 232 \cdot 24 \sqrt{ 7 } $$$$ L = 5568 \sqrt{ 7 } $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 13456 $ and $ L = 5568 \sqrt{ 7 } $ we have:

$$ A = 13456 + 5568 \sqrt{ 7 } $$ $$ A = 13456 + 5568 \sqrt{ 7 } $$ $$ A = 28187.5433 $$

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