Find the circumcircle radius $ R $ of an equilateral triangle if altitude $h = 23$.

$R = \frac{ 46 }{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ h = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$

After substituting $ h = 23 $ we have:

$$ 23 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 23 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 46 $$ $$ a = \dfrac{ 46 }{ \sqrt{ 3 } } $$ $$ a = \frac{ 46 \sqrt{ 3}}{ 3 } $$

STEP 2: find circumradius $ R $

To find circumradius $ R $ use formula:

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

After substituting $ a = \frac{ 46 \sqrt{ 3}}{ 3 } $ we have:

$$ R = \dfrac{ \sqrt{ 3 } \cdot \frac{ 46 \sqrt{ 3}}{ 3 } }{ 3 } $$$$ R = \dfrac{ 46 }{ 3 } $$ $$ R = \frac{ 46 }{ 3 } $$

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