STEP 1: find base diagonal $ d $

To find base diagonal $ d $ use formula:

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

After substituting $a = 10\, \text{cm}$ we have:

$$ d = \sqrt{ 2 } \cdot 10\, \text{cm} $$ $$ d = 10 \sqrt{ 2 }\, \text{cm} $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $V = 500\, \text{cm}$ and $a = 10\, \text{cm}$ we have:

$$ 500\, \text{cm} = \dfrac{ 10\, \text{cm} ^{ 2 } \cdot h }{ 3 } $$$$ 500\, \text{cm} \cdot 3 = 10\, \text{cm} ^{ 2 } \cdot h $$$$ 1500\, \text{cm} = 100\, \text{cm}^2 \cdot h $$$$ h = \dfrac{ 1500\, \text{cm} }{ 100\, \text{cm}^2 } $$$$ h = 15\, \text{cm}^-1 $$

STEP 3: find lateral edge $ e $

To find lateral edge $ e $ use Pythagorean Theorem:

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

After substituting $h = 15\, \text{cm}^-1$ and $d = 10 \sqrt{ 2 }\, \text{cm}$ we have:

$$ \left( 15\, \text{cm}^-1 \right)^{2} + \frac{ \left( 10 \sqrt{ 2 }\, \text{cm} \right)^{2} }{ 4 }= e^2 $$ $$ 225\, \text{cm}^-2 + \frac{ 200\, \text{cm}^2 }{ 4 }= e^2 $$ $$ 225\, \text{cm}^-2 + 50\, \text{cm}^2 = e^2 $$ $$ e^2 = 275\, \text{cm}^-2 $$ $$ e = \sqrt{ 275\, \text{cm}^-2 } $$$$ e = 5 \sqrt{ 11 }\, \text{cm}^-1 $$

Pyramid Calculator