Find the volume $ V $ of a pyramid if height $h = 100000$ and lateral edge $e = 116000$.

$V = 2.304E+14$

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 = 100000 $ and $ e = 116000 $ we have:

$$ 100000^2 + \frac{ d^2 }{ 4 } = 116000^2 $$ $$ \frac{ d^2 }{ 4 } = 116000^2 - 100000^2 $$ $$ \frac{ d^2 }{ 4 } = 13456000000 - 10000000000 $$ $$ d^2 = 3456000000 \cdot 4 $$ $$ d^2 = 13824000000 $$ $$ d = \sqrt{ 13824000000 } $$$$ d = 48000 \sqrt{ 6 } $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d = 48000 \sqrt{ 6 } $ we have:

$$ 48000 \sqrt{ 6 } = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 48000 \sqrt{ 6 } }{ \sqrt{ 2 } } $$ $$ a = 48000 \sqrt{ 3 } $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 48000 \sqrt{ 3 } $ and $ h = 100000 $ we have:

$$ V = \dfrac{ 6912000000 \cdot 100000 }{ 3 }$$$$ V = \dfrac{ 6.912E+14 }{ 3 } $$$$ V = 2.304E+14 $$

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