Find the area $ A $ of a rhombus if diagonal $d_1 = 10$ and height $h = 6$.

$A = 37.5$

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 $ and $ d_1 = 10 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 6 }{ 10 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 3 }{ 5 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 3 }{ 5 } \right) $$ $$ \frac{ \alpha }{ 2 } = 36.8699^o $$$$ \alpha = 36.8699^o \cdot 2 $$$$ \alpha = 73.7398^o $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ \alpha = 73.7398^o $ and $ h = 6 $ we have:

$$ \sin( 73.7398^o ) = \dfrac{ 6 }{ a } $$ $$ 0.96 = \dfrac{ 6 }{ a } $$ $$ a = \dfrac{ 6 }{ 0.96 } $$ $$ a = 6.25 $$

STEP 3: find area $ A $

To find area $ A $ use formula:

$$ A = a \cdot h $$

After substituting $ a = 6.25 $ and $ h = 6 $ we have:

$$ A = 6.25 \cdot 6 $$$$ A = 6.25 \cdot 6 $$$$ A = 37.5 $$

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