Find the volume $ V $ of a pyramid if side $a = 12$ and lateral edge $e = 18$.

$V = 288 \sqrt{ 7 }$

STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $ a = 12 $ we have:

$$ d = \sqrt{ 2 } \cdot 12 $$ $$ d = 12 \sqrt{ 2 } $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ d = 12 \sqrt{ 2 } $ and $ e = 18 $ we have:

$$ h ^ {\,2} + \frac{ \left(12 \sqrt{ 2 }\right)^2 }{ 4 } = 18^2 $$ $$ h ^ {\,2} + \frac{ 288 }{ 4 } = 18^2 $$ $$ h ^ {\,2} + 72 = 18^2 $$ $$ h ^ {\,2} = 324 - 72 $$ $$ h ^ {\,2} = 252 $$ $$ h = \sqrt{ 252 } $$$$ h = 6 \sqrt{ 7 } $$

STEP 3: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 12 $ and $ h = 6 \sqrt{ 7 } $ we have:

$$ V = \dfrac{ 144 \cdot 6 \sqrt{ 7 } }{ 3 }$$$$ V = \dfrac{ 864 \sqrt{ 7 } }{ 3 } $$$$ V = 288 \sqrt{ 7 } $$

Pyramid Calculator