Find the area $ A $ of an equilateral triangle if incircle radius $r = 8$.

$A = 192 \sqrt{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ r = 8 $ we have:

$$ 8 = \dfrac{ \sqrt{ 3 } \cdot a }{ 6 } $$ $$ \sqrt{ 3 } \cdot a = 8 \cdot 6 $$ $$ \sqrt{ 3 } \cdot a = 48 $$ $$ a = \dfrac{ 48 }{ \sqrt{ 3 } } $$ $$ a = 16 \sqrt{ 3 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 16 \sqrt{ 3 } $ we have:

$$ A = \dfrac{ \sqrt{ 3 } \cdot \left(16 \sqrt{ 3 }\right)^2 }{ 4 } $$ $$ A = \dfrac{ \sqrt{ 3 } \cdot 768 }{ 4 }$$ $$ A = \dfrac{ 768 \sqrt{ 3 } }{ 4 }$$ $$ A = 192 \sqrt{ 3 } $$

Equilateral Triangle Calculator