Find the angle $ \alpha $ of a rhombus if side $a = 84$ and diagonal $d_2 = 22$.

$\alpha = 15.1807^o$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $ d_2 = 22 $ and $ a = 84 $ we have:

$$ d_1 ^ {\,2} + 22^2 = 4 \cdot 84^2 $$ $$ d_1 ^ {\,2} + 484 = 28224 $$ $$ d_1 ^ {\,2} = 28224 - 484 $$ $$ d_1 ^ {\,2} = 27740 $$ $$ d_1 = \sqrt{ 27740 } $$$$ d_1 = 2 \sqrt{ 6935 } $$

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 = 22 $ and $ d_1 = 2 \sqrt{ 6935 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 22 }{ 2 \sqrt{ 6935 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 11 \sqrt{ 6935}}{ 6935 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 11 \sqrt{ 6935}}{ 6935 } \right) $$ $$ \frac{ \alpha }{ 2 } = 7.5904^o $$$$ \alpha = 7.5904^o \cdot 2 $$$$ \alpha = 15.1807^o $$

Rhombus Calculator