Find the area $ A $ of an equilateral triangle if circumcircle radius $R = 8$.

$A = 48 \sqrt{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ R = 8 $ we have:

$$ 8 = \dfrac{ \sqrt{ 3 } \cdot a }{ 3 } $$ $$ \sqrt{ 3 } \cdot a = 8 \cdot 3 $$ $$ \sqrt{ 3 } \cdot a = 24 $$ $$ a = \dfrac{ 24 }{ \sqrt{ 3 } } $$ $$ a = 8 \sqrt{ 3 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 8 \sqrt{ 3 } $ we have:

$$ A = \dfrac{ \sqrt{ 3 } \cdot \left(8 \sqrt{ 3 }\right)^2 }{ 4 } $$ $$ A = \dfrac{ \sqrt{ 3 } \cdot 192 }{ 4 }$$ $$ A = \dfrac{ 192 \sqrt{ 3 } }{ 4 }$$ $$ A = 48 \sqrt{ 3 } $$

Equilateral Triangle Calculator