# 11/11/11 — Equilateral Triangle Day!

While others celebrate the number 11 on this special day, I prefer to honor the Equilateral Triangle.

Last year, on 10/10/10, I celebrated the symmetry of the equilateral triangle. This year, I offer a favorite Proof Without Words. Well, a proof with *some* words. In any event, we will use equilateral triangles to prove that the following infinite series

+ + + + . . .

is .

Consider the following diagram.

Notice that the largest blue equilateral triangle is the area of the entire equilateral triangle. The next largest blue triangle is of , or of the entire triangle. The next largest blue triangle is of the original triangle, and so on.

So, the sum of the blue triangles is

+ + + + . . .

Let’s call this *S*.

Now, here’s the magic: the sum of the red triangles is also *S*! This is true because for every blue triangle, there is a congruent red triangle right next to it. Similarly, the sum of the yellow triangle is also *S*.

When you put all the blue, red, and yellow triangles together, you get the original triangle, whose area is 1. Thus, *3S = 1*, and so

Therefore, we have

+ + + + . . . =

Happy Equilateral Triangle Day!

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## 5 Comments

## Whered · November 11, 2011 at 9:32 am

What about Veteran’s day?

## MrHonner · November 11, 2011 at 9:45 am

I recognize Armistice Day, but the mathematics that naturally comes to mind (total losses due to war) is too much of a downer to write about.

## janinelson (@janinelson) · November 11, 2011 at 9:37 am

Thanks… I love this one… only wish we were in school today– I would use it with my students!! Oh well, guess I will need to wait until next year 12/12/12

## MrHonner · November 11, 2011 at 9:46 am

I love this one, too–a student showed this to me last year.

And make sure to use it on 12/12/12! Otherwise, you’ll be waiting a

longtime for the next opportunity.## chrisharrow · November 11, 2011 at 10:49 am

I think the image is gorgeous as a proof without words.