Find the angle $ \alpha $ of a rhombus if side $a = 8$ and diagonal $d_2 = \frac{ 113 }{ 10 }$.

$\alpha = 172.028^o$

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 = \frac{ 113 }{ 10 } $ and $ a = 8 $ we have:

$$ d_1 ^ {\,2} + \left(\frac{ 113 }{ 10 }\right)^2 = 4 \cdot 8^2 $$ $$ d_1 ^ {\,2} + \frac{ 12769 }{ 100 } = 256 $$ $$ d_1 ^ {\,2} = 256 - \frac{ 12769 }{ 100 } $$ $$ d_1 ^ {\,2} = \frac{ 12831 }{ 100 } $$ $$ d_1 = \sqrt{ \frac{ 12831 }{ 100 } } $$$$ d_1 = \frac{\sqrt{ 12831 }}{ 10 } $$

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 = \frac{ 113 }{ 10 } $ and $ d_1 = \frac{\sqrt{ 12831 }}{ 10 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ \frac{ 113 }{ 10 } }{ \frac{\sqrt{ 12831 }}{ 10 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 113 \sqrt{ 12831}}{ 12831 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 113 \sqrt{ 12831}}{ 12831 } \right) $$ $$ \frac{ \alpha }{ 2 } = 86.014^o $$$$ \alpha = 86.014^o \cdot 2 $$$$ \alpha = 172.028^o $$

Rhombus Calculator