STEP 1: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $A = \sqrt{ 3 }\, \text{cm}$ and $a = \sqrt{ 2 }\, \text{cm}$ we have:

$$ \sqrt{ 3 }\, \text{cm} = \sqrt{ 2 }\, \text{cm} \cdot h $$$$ h = \dfrac{ \sqrt{ 3 }\, \text{cm} }{ \sqrt{ 2 }\, \text{cm} } $$$$ h = \frac{\sqrt{ 6 }}{ 2 } $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \alpha \right) = \dfrac{ h }{ a } $$

After substituting $h = \dfrac{\sqrt{ 6 }}{ 2 }\, \text{cm}^0$ and $a = \sqrt{ 2 }\, \text{cm}$ we have:

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

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $\alpha = 60^o$ and $h = \dfrac{\sqrt{ 6 }}{ 2 }\, \text{cm}^0$ we have:

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

Rhombus Calculator