Find the angle $ \alpha $ of a rhombus if area $A = 20449$ and diagonal $d_1 = 30$.

Problem has no solutions.

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 20449 $ and $ d_1 = 30 $ we have:

$$ 20449 = \dfrac{ 30 \cdot d_2 }{ 2 } $$$$ 20449 \cdot 2 = 30 \cdot d_2 $$$$ 40898 = 30 \cdot d_2 $$$$ d_2 = \dfrac{ 40898 }{ 30 } $$$$ d_2 = \frac{ 20449 }{ 15 } $$

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 = \frac{ 20449 }{ 15 } $ and $ d_1 = 30 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 20449 }{ 15 } }{ 30 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 20449 }{ 450 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 20449 }{ 450 } \right) $$

$ \arcsin(45.442) $ is not defined $ \Longrightarrow $ The problem has no solution.

Rhombus Calculator