Find the side $ a $ of a rhombus if diagonal $d_1 = 6$ and angle $\alpha = 60^o$.

$a = 2 \sqrt{ 3 }$

STEP 1: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 30^o $ and $ d_1 = 6 $ we have:

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

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ \alpha = 60^o $ and $ h = 3 $ we have:

$$ \sin( 60^o ) = \dfrac{ 3 }{ a } $$ $$ \frac{\sqrt{ 3 }}{ 2 } = \dfrac{ 3 }{ a } $$ $$ a = \dfrac{ 3 }{ \frac{\sqrt{ 3 }}{ 2 } } $$ $$ a = 2 \sqrt{ 3 } $$

Rhombus Calculator