Find the area $ A $ of an equilateral triangle if altitude $h = 9$.

$A = 27 \sqrt{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ h = 9 $ we have:

$$ 9 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 9 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 18 $$ $$ a = \dfrac{ 18 }{ \sqrt{ 3 } } $$ $$ a = 6 \sqrt{ 3 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 6 \sqrt{ 3 } $ we have:

$$ A = \dfrac{ \sqrt{ 3 } \cdot \left(6 \sqrt{ 3 }\right)^2 }{ 4 } $$ $$ A = \dfrac{ \sqrt{ 3 } \cdot 108 }{ 4 }$$ $$ A = \dfrac{ 108 \sqrt{ 3 } }{ 4 }$$ $$ A = 27 \sqrt{ 3 } $$

Equilateral Triangle Calculator