## Which Triangle is More Equilateral? Part II

A surprising amount of interest was generated by my question “Which Triangle is More Equilateral?” With the passing of consecutive **Isosceles Triangle days**, I wondered: which triangle was *more equilateral,* the 10-10-11 triangle, or the 10-11-11 triangle?

Many rich and interesting conversations arose as a result of this question. Colleagues, students, and commenters on the original post offered good ideas about how to approach both the question itself and the concept in general.

There is a lot to say about this seemingly simple problem, but I’ll begin by sharing my approach to the question “Which triangle is more equilateral?”

After playing around with several ideas, I tried to get to the heart of *equilateralness*. **Equilateral** is a well-defined idea (all sides congruent), but how could we relax that definition and quanitfy **equilateralness **in a continuous, rather than a discrete, way?

I chose to to think of *equilateralness *as a measure of how **circle-like** the object is: the more it’s like a circle, the more equilateral that object is. What does it mean to be **circle-like**? I chose the following idea as my foundation: the fundamental characteristic of the circle is that it maximizes area for a given perimeter. From that perspective, I created a measurement.

I define a triangle’s **equilateralness** to be the ratio of its area to the area of the equilateral triangle with the same perimeter. Thus, for triangle ABC with sides *a, b, *and *c*, its **equilateralness **is given by

where [ABC] denotes the area of triangle ABC.

Since the equlateral triangle is *the *triangle of maximum area for a given perimeter, the above measure will be bewtween 0 and 1 for all triangles. The closer its **Eq **is to **1** , the closer the triangle is to being equilateral.

So for the triangles in question, we have

Thus, by a very slim margin, the 10-11-11 triangle is **more equilateral** than the 10-10-11 triangle!

The value in this question is not so much settling on this, or any, particular approach; what’s valuable here is the opportunity to creatively explore a lot of interesting and deep mathematical ideas. I look forward to doing just that!

*Click here to see more in Geometry.*

my brother said that there is a simpler way in doing this

if we have a 10-10-11 triangle it would need to be 10-10-10 to be equilateral

so the triangle would be 10% off from being equilateral

if we have a 10-11-11 triangle it would need to be 11-11-11 to be equilateral

so the triangle would be 1/11 ~ 9% off from being equilateral.

Therefore, the 10-11-11 triangle is “more equilateral”

That’s an interesting approach. What conclusion would be reached if you compared a 9-10-11 triangle and a 9-11-11 triangle in this way?

in response to that my brother said it would be the 9-11-11 triangle.

the 9-11-11 triangle would be 11-11-11 if it were equilateral. that makes the side of 9 to be 2/11 off from being equilateral.

the 9-10-11 triangle would be 10-10-10 if it were equilateral. that makes the side of 9 to be 1/10 off and the side of 11 to also be 1/10 off from being equilateral, adding up to be 2/10 off

that would make the 9-10-11 triangle more off from being equilateral than the 9-11-11 triangle

So that’s what your brother thinks, but what do

youthink?Intuitively I think the 9-10-11 triangle is

moreequilateral. Maybe I should run the above analysis on it and see if the result agrees with my intuition.This idea is entirely fleshed out but what about considering the minimum distance that a vertex would have to displace in order to “make” the triangle equilateral?

This idea kind of breaks down in the sense that a 17-17-16 triangle is just as “almost equilateral” as a 170-170-160 but I’m sure it can be fixed.

That’s a good idea, too, but it gets complicated if you can most efficiently achieve an equilateral triangle by moving more than one vertex.