Find the angle $ \beta $ of a rhombus if diagonal $d_2 = 13542$ and incircle radius $r = 6371$.

$\beta = 140.4148^o$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ h = 2 \cdot r $$

After substituting $ r = 6371 $ we have:

$$ h = 2 \cdot 6371 $$ $$ h = 12742 $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ h = 12742 $ and $ d_2 = 13542 $ we have:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ 12742 }{ 13542 } $$ $$ \sin \left( \frac{ \beta }{ 2 } \right) = \frac{ 6371 }{ 6771 } $$ $$ \frac{ \beta }{ 2 } = \arcsin\left( \frac{ 6371 }{ 6771 } \right) $$ $$ \frac{ \beta }{ 2 } = 70.2074^o $$$$ \beta = 70.2074^o \cdot 2 $$$$ \beta = 140.4148^o $$

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