Find the lateral edge $ e $ of a pyramid if side $a = 41500$ and height $h = 19200$.

$e = 50 \sqrt{ 491906 }$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $ a = 41500 $ we have:

$$ d = \sqrt{ 2 } \cdot 41500 $$ $$ d = 41500 \sqrt{ 2 } $$

STEP 2: find lateral edge $ e $

To find lateral edge $ e $ use Pythagorean Theorem:

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

After substituting $ h = 19200 $ and $ d = 41500 \sqrt{ 2 } $ we have:

$$ 19200^2 + \frac{ \left(41500 \sqrt{ 2 }\right)^2 }{ 4 }= e^2 $$ $$ 368640000 + \frac{ 3444500000 }{ 4 }= e^2 $$ $$ 368640000 + 861125000 = e^2 $$ $$ e^2 = 1229765000 $$ $$ e = \sqrt{ 1229765000 } $$$$ e = 50 \sqrt{ 491906 } $$

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