Find the area $ A $ of an equilateral triangle if circumcircle radius $R = 3$.

$A = \frac{ 27 \sqrt{ 3}}{ 4 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ R = 3 $ we have:

$$ 3 = \dfrac{ \sqrt{ 3 } \cdot a }{ 3 } $$ $$ \sqrt{ 3 } \cdot a = 3 \cdot 3 $$ $$ \sqrt{ 3 } \cdot a = 9 $$ $$ a = \dfrac{ 9 }{ \sqrt{ 3 } } $$ $$ a = 3 \sqrt{ 3 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

$$ A = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 } $$

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

$$ A = \dfrac{ \sqrt{ 3 } \cdot \left(3 \sqrt{ 3 }\right)^2 }{ 4 } $$ $$ A = \dfrac{ \sqrt{ 3 } \cdot 27 }{ 4 }$$ $$ A = \dfrac{ 27 \sqrt{ 3 } }{ 4 }$$ $$ A = \frac{ 27 \sqrt{ 3}}{ 4 } $$

Equilateral Triangle Calculator