Find the diagonal $ d_1 $ of a rhombus if side $a = \sqrt{ 2 }$ and area $A = \sqrt{ 3 }$.

$d_1 = \sqrt{ 6 }$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

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

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

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{\sqrt{ 6 }}{ 2 } $ and $ a = \sqrt{ 2 } $ we have:

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

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 30 $ and $ h = \frac{\sqrt{ 6 }}{ 2 } $ we have:

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

Rhombus Calculator