# Discovered a neat formula (equilateral triangles)

## Main Question or Discussion Point

If you use this formula: $s^2$$\times$$\sqrt{3}$$\div4$ it gives you the area of said equilateral triangle.

I guess it's pretty neat i discovered it, but it doesn't seem very useful overall, anyone know any other uses for it?

Mark44
Mentor
If you use this formula: $s^2$$\times$$\sqrt{3}$$\div4$ it gives you the area of said equilateral triangle.

I guess it's pretty neat i discovered it, but it doesn't seem very useful overall, anyone know any other uses for it?
Not to diminish your discovery in any way, but that formula has been known for a long time. For an equilateral triangle, all three interior angles are 60°. If you orient the triangle so that one of the sides is horizontal, the bisector of the angle at the top divides the base into two equal segments, and divides the triangle into two congruent 30° - 60° right triangles. The base of each of these smaller triangles is s/2. Using the theorem of Pythagoras or trig, one can find that the shared vertical side of the two triangles is s * $\sqrt{3}/2$. Having found this information, the area of the larger triangle is (1/2)*base*height = $(s/2) *s *\sqrt{3}/2$, same as what you discovered..

jbunniii
Homework Helper
Gold Member
I'm not sure what other uses you envision aside from calculating the area of an equilateral triangle. You can use it as a building block to find the areas of objects that can decomposed into equilateral triangles, for example a regular hexagon with side $s$ can be broken into six equilateral triangles of side $s$, so its area is $3\sqrt{3}s^2/2$.

Mentallic
Homework Helper
Let's not crush shadowboy's accomplishment all at once now.

I remember when my class was learning about Pythagoras' theorem, and soon later I figured out the distance from one corner of a cube to the opposite corner. I felt like I had stumbled onto something brilliant.

But of course, my teacher felt the need to slap some sense into me, and from that day on I never again felt like any of my personal discoveries were feats of genius.

It didn't stop me from searching for new neat little formulas though! Keep it up shadowboy13.