Find the diagonal $ d_1 $ of a rhombus if side $a = 8$ and height $h = \frac{ 693 }{ 100 }$.

$d_1 = 13.8546$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = \frac{ 693 }{ 100 } $ and $ a = 8 $ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 693 }{ 100 } }{ 8 } $$ $$ \sin \left( \alpha \right) = \frac{ 693 }{ 800 } $$ $$ \alpha = \arcsin\left( \frac{ 693 }{ 800 } \right) $$ $$ \alpha = 60.0257^o $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 30.0129 $ and $ h = \frac{ 693 }{ 100 } $ we have:

$$ \sin \left( \frac{ 60.0257^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( 30.0129 ) = \dfrac{ \frac{ 693 }{ 100 } }{ d_1 } $$ $$ 0.5002 = \dfrac{ \frac{ 693 }{ 100 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 693 }{ 100 } }{ 0.5002 } $$ $$ d_1 = 13.8546 $$

Rhombus Calculator