Find the area $ A $ of a rhombus if diagonal $d_1 = 15$ and height $h = 12$.

$A = 150$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ h = 12 $ and $ d_1 = 15 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 12 }{ 15 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 4 }{ 5 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 4 }{ 5 } \right) $$ $$ \frac{ \alpha }{ 2 } = 53.1301^o $$$$ \alpha = 53.1301^o \cdot 2 $$$$ \alpha = 106.2602^o $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ \alpha = 106.2602^o $ and $ h = 12 $ we have:

$$ \sin( 106.2602^o ) = \dfrac{ 12 }{ a } $$ $$ 0.96 = \dfrac{ 12 }{ a } $$ $$ a = \dfrac{ 12 }{ 0.96 } $$ $$ a = 12.5 $$

STEP 3: find area $ A $

To find area $ A $ use formula:

$$ A = a \cdot h $$

After substituting $ a = 12.5 $ and $ h = 12 $ we have:

$$ A = 12.5 \cdot 12 $$$$ A = 12.5 \cdot 12 $$$$ A = 150 $$

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