# Another Equilateral Comparison

The passing of consecutive isosceles triangle days has me once again thinking about the question “Which Triangle is More Equilateral?”

I first considered the question on 10/10/11, comparing the 10-10-11 triangle and the 10-11-11 triangle. After a spirited discussion, I offered one approach to the question here. The problem gave me lots to think about, both mathematically and pedagogically, and I reflected on what I liked about this problem here.

But as 12/11/11 and 12/12/11 pass, I thought I’d revisit my strategy for answering the question “Which triangle is more equilateral?”

My basic strategy, outlined in more detail here, is to ultimately to quantify the *circleness* of each triangle. To me, being equilateral is all about trying to be as much like a circle as possible. So I created a measure to determine how close to circlehood a triangle is. Here are the numbers.

The 11-12-12 triangle’s measure is closer to 1, thus making it the more equilateral triangle.

**Related Posts**

- Which Triangle is More Equilateral?
- Which Triangle is More Equilateral? Part II
- Which Triangle is More Equilateral? 2012 Edition
- An Ode to Equilateralism

## 2 Comments

## Roy Wright · December 12, 2011 at 10:47 am

Good stuff. I think this sort of thinking — inventing a metric to quantify something where there are multiple reasonable possibilities — is both a valuable skill in modern life, and a potentially fertile ground for developing deep mathematical understanding (along with creativity). But like just about any topic in mathematical science that is open to personal preference, discussion, and trade-offs, it’s heavily underemphasized. It’s exciting to see, in the comments on your October post, alternative metrics being introduced and discussed, especially when they are seemingly simpler but make less aesthetic sense.

## MrHonner · December 12, 2011 at 6:47 pm

You have enumerated many of the virtues of this problem, which I also appreciated myself (I discussed these, and others, here).

The creativity inherent in all mathematical processes is highly under-emphasized at all levels. And it’s such a draw for students!