Find the area $ A $ of a pyramid if side $a = 5$ and lateral edge $e = 8$.

$A = 100.9934$

STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 5 $ we have:

$$ B = 5^2 $$ $$ B = 25 $$

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 = 5 $ and $ e = 8 $ we have:

$$ s ^ {\,2} + \frac{ 5^2 }{ 4 } = 8^2 $$ $$ s ^ {\,2} + \frac{ 25 }{ 4 } = 8^2 $$ $$ s ^ {\,2} + \frac{ 25 }{ 4 } = 8^2 $$ $$ s ^ {\,2} = 64 - \frac{ 25 }{ 4 } $$ $$ s ^ {\,2} = \frac{ 231 }{ 4 } $$ $$ s = \sqrt{ \frac{ 231 }{ 4 } } $$$$ s = \frac{\sqrt{ 231 }}{ 2 } $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 5 $ and $ s = \frac{\sqrt{ 231 }}{ 2 } $ we have:

$$ L = 10 \cdot \frac{\sqrt{ 231 }}{ 2 } $$$$ L = 5 \sqrt{ 231 } $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 25 $ and $ L = 5 \sqrt{ 231 } $ we have:

$$ A = 25 + 5 \sqrt{ 231 } $$ $$ A = 25 + 5 \sqrt{ 231 } $$ $$ A = 100.9934 $$

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