STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ h }{ d_1 } $$

After substituting $h = 6 \sqrt{ 3 }\, \text{cm}$ and $d_1 = 12 \sqrt{ 3 }\, \text{cm}$ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 6 \sqrt{ 3 }\, \text{cm} }{ 12 \sqrt{ 3 }\, \text{cm} } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 1 }{ 2 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 1 }{ 2 } \right) $$ $$ \frac{ \alpha }{ 2 } = 30^o $$$$ \alpha = 30^o \cdot 2 $$$$ \alpha = 60^o $$

STEP 2: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $\alpha = 60^o$ and $d_1 = 12 \sqrt{ 3 }\, \text{cm}$ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$ $$ \sin( 30 ) = \dfrac{ d_2 }{ 12 \sqrt{ 3 } } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ d_2 }{ 12 \sqrt{ 3 } } $$$$ d_2 = \frac{ 1 }{ 2 } \cdot 12 \sqrt{ 3 } $$$$ d_2 = 6 \sqrt{ 3 } $$

Rhombus Calculator