Find the volume $ V $ of a pyramid if lateral edge $e = 10$ and base area $B = 6$.

$V = 2 \sqrt{ 97 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

After substituting $ B = 6 $ we have:

$$ B = a^2 $$ $$ a^2 = 6 $$ $$ a = \sqrt{ 6 } $$

STEP 2: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

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

$$ d = \sqrt{ 2 } \cdot \sqrt{ 6 } $$ $$ d = 2 \sqrt{ 3 } $$

STEP 3: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 2 \sqrt{ 3 } $ and $ e = 10 $ we have:

$$ h ^ {\,2} + \frac{ \left(2 \sqrt{ 3 }\right)^2 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + \frac{ 12 }{ 4 } = 10^2 $$ $$ h ^ {\,2} + 3 = 10^2 $$ $$ h ^ {\,2} = 100 - 3 $$ $$ h ^ {\,2} = 97 $$ $$ h = \sqrt{ 97 } $$

STEP 4: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = \sqrt{ 6 } $ and $ h = \sqrt{ 97 } $ we have:

$$ V = \dfrac{ 6 \cdot \sqrt{ 97 } }{ 3 }$$$$ V = \dfrac{ 6 \sqrt{ 97 } }{ 3 } $$$$ V = 2 \sqrt{ 97 } $$

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