Find the angle $ \alpha $ of a rhombus if area $A = 150$ and diagonal $d_1 = 30$.

$\alpha = 38.9424^o$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ A = \dfrac{ d_1 \cdot d_2 }{ 2 } $$

After substituting $ A = 150 $ and $ d_1 = 30 $ we have:

$$ 150 = \dfrac{ 30 \cdot d_2 }{ 2 } $$$$ 150 \cdot 2 = 30 \cdot d_2 $$$$ 300 = 30 \cdot d_2 $$$$ d_2 = \dfrac{ 300 }{ 30 } $$$$ d_2 = 10 $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ d_2 = 10 $ and $ d_1 = 30 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 10 }{ 30 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 1 }{ 3 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 1 }{ 3 } \right) $$ $$ \frac{ \alpha }{ 2 } = 19.4712^o $$$$ \alpha = 19.4712^o \cdot 2 $$$$ \alpha = 38.9424^o $$

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