Question

How do you derive the formula for the area of an equilateral triangle?

Answer 1

Area of an equilateral triangle with sides of length `s`
`color(white)("XXX")=sqrt(3)/4*s^2`
(Derivation below)

Explanation:

The common formula for the area of a triangle is
`color(white)("XXX")"Area"_triangle = ("base" xx "height")/2`

For an equilateral triangle with sides of length `s`

`"base" = s`

and the `"height"` can be calculated using the Pythagorean Theorem as:

`height" = sqrt(s^2-(s/2)^2)`

`color(white)("XXXXX")=sqrt((3s^2)/4)`

`color(white)("XXXXX")=s/2sqrt(3)`

and therefore the area is
`color(white)("XXX")"Area"_triangle = s xx (s/2sqrt(3))/2`

`color(white)("XXXXXXX")=sqrt(3)/4 s^2`