Question

show that the area of an equilateral triangle is under root 3 upon 4 x square, where side is x

Answer 1

Now apply the Pythagorean theorem to get the height (h) or the length of the line you see in red


a2 = (a/2)2 + h2

a2 = a2/4 + h2

a2 − a2/4 = h2

4a2/4 − a2/4 = h2

3a2/4 = h2

h = √(3a2/4) 

h = (√(3)×a)/2

Area = (base × h)/2 

base × h = (a × √(3)×a)/2 = (a2× √(3))/2

Dividing by 2 is the same as multiplying the denominator by 2. Therefore, the formula is



Answer 2

let, all sides of an equilateral triangle be x
We Know To Calculate The Area Of Triangle By Heron's Formula

Here semiperimeter= x+x+x/ 2 = 3x/2

s-x = 3x/2-x= x/2

Putting These Values In the formula

area of the triangle
A = Root Over ( s(s-a)(s-b)(s-c) )

here a = b = c = a=x let,

Hence

A = Root Over (3x/2*x/2*x/2*x/2)

A = √ (3x^4/16)

A = x^2/4 * √ 3

A = √3 *x^2 /4