Area of an Equilateral Triangle

Find the area of an equilateral triangle of side a.

SOLUTION:

Using Heron’s Formula, the formula of the triangle is given by

 A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}

where s is the semi-perimeter, s=\frac{a+b+c}{2}

Since it is an equilateral triangle, a=b=c.

s=\frac{a+a+a}{2}=\frac{3a}{2}

Threfore, the area is 

A=\sqrt{\frac{3a}{2}\left(\frac{3a}{2}-a\right)^3}

A=\sqrt{\frac{3a}{2}\left(\frac{a}{2}\right)^3}

A=\sqrt{\frac{3a}{2}\left(\frac{a^3}{8}\right)}

A=\sqrt{\frac{3a^4}{16}}

A=\frac{\sqrt{3}}{4}a^2

 

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