STEP 1: find side $ a $

To find side $ a $ use formula:

$$ h = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$

After substituting $h = 13\, \text{cm}$ we have:

$$ 13\, \text{cm} = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 13\, \text{cm} \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 26\, \text{cm} $$ $$ a = \dfrac{ 26\, \text{cm} }{ \sqrt{ 3 } } $$ $$ a = \frac{ 26 \sqrt{ 3}}{ 3 }\, \text{cm} $$

STEP 2: find length $ l $

To find length $ l $ use formula:

$$ V = \dfrac{ \sqrt{ 3 } \cdot a ^{ 2 } \cdot l }{ 4 } $$

After substituting $V = 3640\, \text{cm}$ and $a = \dfrac{ 26 \sqrt{ 3}}{ 3 }\, \text{cm}$ we have:

$$ 3640\, \text{cm} = \dfrac{ \sqrt{ 3 } \cdot \frac{ 26 \sqrt{ 3}}{ 3 }\, \text{cm} ^{ 2 } \cdot l }{ 4 } $$$$ 3640\, \text{cm} \cdot 4 = \sqrt{ 3 } \cdot \frac{ 26 \sqrt{ 3}}{ 3 }\, \text{cm} ^{ 2 } \cdot l $$$$ 14560\, \text{cm} = \sqrt{ 3 } \cdot \frac{ 26 \sqrt{ 3}}{ 3 }\, \text{cm} ^{ 2 } \cdot l $$$$ 14560\, \text{cm} = \frac{ 676 \sqrt{ 3}}{ 3 }\, \text{cm}^2 \cdot l $$$$ l = \dfrac{ 14560\, \text{cm} }{ \frac{ 676 \sqrt{ 3}}{ 3 }\, \text{cm}^2 } $$$$ l = \frac{ 280 \sqrt{ 3}}{ 13 }\, \text{cm}^-1 $$

Triangular Prism Calculator