Find the lateral edge $ e $ of a pyramid if side $a = 15$ and volume $V = 900$.

$e = \frac{ 3 \sqrt{ 114}}{ 2 }$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $ a = 15 $ we have:

$$ d = \sqrt{ 2 } \cdot 15 $$ $$ d = 15 \sqrt{ 2 } $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ V = 900 $ and $ a = 15 $ we have:

$$ 900 = \dfrac{ 15 ^{ 2 } \cdot h }{ 3 } $$$$ 900 \cdot 3 = 15 ^{ 2 } \cdot h $$$$ 2700 = 225 \cdot h $$$$ h = \dfrac{ 2700 }{ 225 } $$$$ h = 12 $$

STEP 3: find lateral edge $ e $

To find lateral edge $ e $ use Pythagorean Theorem:

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

After substituting $ h = 12 $ and $ d = 15 \sqrt{ 2 } $ we have:

$$ 12^2 + \frac{ \left(15 \sqrt{ 2 }\right)^2 }{ 4 }= e^2 $$ $$ 144 + \frac{ 450 }{ 4 }= e^2 $$ $$ 144 + \frac{ 225 }{ 2 } = e^2 $$ $$ e^2 = \frac{ 513 }{ 2 } $$ $$ e = \sqrt{ \frac{ 513 }{ 2 } } $$$$ e = \frac{ 3 \sqrt{ 114}}{ 2 } $$

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