STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $a = 6\, \text{cm}$ we have:

$$ d = \sqrt{ 2 } \cdot 6\, \text{cm} $$ $$ d = 6 \sqrt{ 2 }\, \text{cm} $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $d = 6 \sqrt{ 2 }\, \text{cm}$ and $e = 12\, \text{cm}$ we have:

$$ h ^ {\,2} + \frac{ \left( 6 \sqrt{ 2 }\, \text{cm} \right)^{2} }{ 4 } = \left( 12\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + \frac{ 72\, \text{cm}^2 }{ 4 } = \left( 12\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + 18\, \text{cm}^2 = \left( 12\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} = 144\, \text{cm}^2 - 18\, \text{cm}^2 $$ $$ h ^ {\,2} = 126\, \text{cm}^2 $$ $$ h = \sqrt{ 126\, \text{cm}^2 } $$$$ h = 3 \sqrt{ 14 }\, \text{cm} $$

Pyramid Calculator