Question

What is area of equilateral triangle?

Answer 1

If the sides of an equilateral triangle are all of length `a`, then the area is `sqrt(3)/4a^2`

Explanation:

Consider an equilateral triangle with sides of length `a`.

If you bisect it to make two right angled triangles, then those triangles will have hypotenuse of length `a`, shortest side of length `a/2` and other side of length:

`sqrt(a^2-(a/2)^2) = sqrt(a^2-a^2/4) = sqrt((3a^2)/4) = (sqrt(3)a)/2`

The two right angled triangles can be rearranged (turning one over) into a rectangle with sides `(sqrt(3)a)/2` and `a/2`.

The area of the rectangle, which is the same as the area of the original triangle is:

`(sqrt(3)a)/2*a/2 = sqrt(3)/4a^2`