Find the side $ a $ of a pyramid if height $h = 5$ and lateral edge $e = 15$.

$a = 20$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use Pythagorean Theorem:

$$ h^2 + \frac{ d^2 }{ 4 } = e^2 $$

After substituting $ h = 5 $ and $ e = 15 $ we have:

$$ 5^2 + \frac{ d^2 }{ 4 } = 15^2 $$ $$ \frac{ d^2 }{ 4 } = 15^2 - 5^2 $$ $$ \frac{ d^2 }{ 4 } = 225 - 25 $$ $$ d^2 = 200 \cdot 4 $$ $$ d^2 = 800 $$ $$ d = \sqrt{ 800 } $$$$ d = 20 \sqrt{ 2 } $$

STEP 2: find side $ a $

To find side $ a $ use formula:

$$ d = \sqrt{ 2 } \cdot a $$

After substituting $ d = 20 \sqrt{ 2 } $ we have:

$$ 20 \sqrt{ 2 } = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 20 \sqrt{ 2 } }{ \sqrt{ 2 } } $$ $$ a = 20 $$

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