Find the volume $ V $ of a pyramid if side $a = 6$ and lateral edge $e = 7.5$.

$V = 74.2159$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $ a = 6 $ we have:

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

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 6 \sqrt{ 2 } $ and $ e = 7.5 $ we have:

$$ h ^ {\,2} + \frac{ \left(6 \sqrt{ 2 }\right)^2 }{ 4 } = 7.5^2 $$ $$ h ^ {\,2} + \frac{ 72 }{ 4 } = 7.5^2 $$ $$ h ^ {\,2} + 18 = 7.5^2 $$ $$ h ^ {\,2} = 56.25 - 18 $$ $$ h ^ {\,2} = 38.25 $$ $$ h = \sqrt{ 38.25 } $$$$ h = 6.1847 $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 6 $ and $ h = 6.1847 $ we have:

$$ V = \dfrac{ 36 \cdot 6.1847 }{ 3 }$$$$ V = \dfrac{ 222.6477 }{ 3 } $$$$ V = 74.2159 $$

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