Find the incircle radius $ r $ of an equilateral triangle if circumcircle radius $R = \frac{ 53 }{ 25 }$.

$r = \frac{ 53 }{ 50 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ R = \dfrac{ \sqrt{ 3 } \cdot a }{ 3 } $$

After substituting $ R = \frac{ 53 }{ 25 } $ we have:

$$ \frac{ 53 }{ 25 } = \dfrac{ \sqrt{ 3 } \cdot a }{ 3 } $$ $$ \sqrt{ 3 } \cdot a = \frac{ 53 }{ 25 } \cdot 3 $$ $$ \sqrt{ 3 } \cdot a = \frac{ 159 }{ 25 } $$ $$ a = \dfrac{ \frac{ 159 }{ 25 } }{ \sqrt{ 3 } } $$ $$ a = \frac{ 53 \sqrt{ 3}}{ 25 } $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

$$ r = \dfrac{ \sqrt{ 3 } \cdot a }{ 6 } $$

After substituting $ a = \frac{ 53 \sqrt{ 3}}{ 25 } $ we have:

$$ r = \dfrac{ \sqrt{ 3 } \cdot \frac{ 53 \sqrt{ 3}}{ 25 } }{ 6 } $$$$ r = \dfrac{ \frac{ 159 }{ 25 } }{ 6 } $$ $$ r = \frac{ 53 }{ 50 } $$

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