Theory:

Area of an equilateral triangle using Heron's formula:
equilateral triangle.png
 
\(AB\) \(=\) \(BC\) \(=\) \(CA\) \(=\) \(a\)
 
That is, \(a\) \(=\) \(b\) \(=\) \(c\)
 
s=a+b+c2
 
\(s\) \(=\) a+a+a2 \(=\) 3a2
 
Area of a triangle \(=\) ssasbsc
 
\(=\) 3a23a2a3a2a3a2a
 
\(=\) 3a23a2a23a2a23a2a2
 
\(=\) 3a2a2a2a2
 
\(=\) 3×a22×a22
 
\(=\) a2×a23
 
\(=\) 34a2
 
Therefore, the area of an equilateral triangle \(=\) 34a2 sq. units.
Example:
Calculate the area of an equilateral triangle of side \(4 \ cm\).
 
\(a\) \(=\) \(b\) \(=\) \(c\) \(=\) \(4 \ cm\)
 
Area of an equilateral triangle \(=\) 34a2
 
\(=\) 34×42
 
\(=\) 34×16
 
\(=\) \(\sqrt{3} \times 4\)
 
\(=\) \(1.73  \times 4\)      [Since \(\sqrt{3} = 1.73\)]
 
\(=\) \(6.92\)
 
Therefore, the area of an equilateral triangle \(=\) \(6.92 \ cm^2\).