Find the circumcircle radius $ R $ of a hexagon if incircle radius $r = \frac{ 9 }{ 16 }$.
Answer
$R = \frac{ 3 \sqrt{ 3}}{ 8 }$
Explanation
STEP 1: find side $ a $
To find side $ a $ use formula:
$$ r = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$After substituting $ r = \frac{ 9 }{ 16 } $ we have:
$$ \frac{ 9 }{ 16 } = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = \frac{ 9 }{ 16 } \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = \frac{ 9 }{ 8 } $$ $$ a = \dfrac{ \frac{ 9 }{ 8 } }{ \sqrt{ 3 } } $$ $$ a = \frac{ 3 \sqrt{ 3}}{ 8 } $$STEP 2:
The radius of the circumcircle is equal the side of the hexagon. $$ R = \frac{ 3 \sqrt{ 3}}{ 8 } $$
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