Question

prove that the area of Equilateral triangle
${3 \div } 4a2$
​

Answer 1

Step-by-step explanation:

Let a triangle, whose sides are a,b,c

since in an equilateral triangle all sides are equal

a=b=c

semi-perimeter of the triangle = (a+b+c)/2

= 3a/2

Using Heron's formulae:

$area \: of \: a \: triangle \: = \sqrt{s(s - a)(s - b)(s - c)} \\ = \sqrt{ \frac{3a}{2} ( \frac{3a}{2} - a) ( \frac{3a}{2} - a) ( \frac{3a}{2} - a)} \: \: \: \: {since \: a = b = c} \\ = \sqrt{ \frac{3a}{2} {( \frac{a}{2} )}^{3} } \\ = \sqrt{ \frac{3a}{2} \times \frac{ {a}^{3} }{8 } } \\ = \sqrt{ \frac{3 {a}^{4} }{16} } \\ = \frac{ \sqrt{3} {a}^{2} }{4}$

PROVED