Is there a measure for how thin or squat a triangle is?

Question

Is there a measure for how thin or squat a triangle is? Similar to eccentricity for ellipses.

Answer 1

I would imagine that the quantity "circumradius/inradus" would be a good candidate. This quantity will be larger for slender triangles.

Intuitively one would expect this to be minimized by an equilateral triangle at a measurement of 2.

Answer 2

Candidate quantities can be found in inequalities for triangles in which equality occurs just for equilateral triangles.

For example, the isoperimetric inequality for triangles asserts that the ratio $p^2/A$ (where $p$ is the perimeter and $A$ the area) is minimal for the equilateral triangle (among triangles). This quantity, then, measures deviation from being equilateral, which is, perhaps, somewhat analogous to eccentricity.

Another example is Jung's inequality, which asserts that the ratio of diameter to circumradius is maximal for the equilateral triangle (among all compact sets, even, not just all triangles; and analogously, the regular simplex in higher dimensions, not just in the plane); this quantity for triangles again measures deviation from being equilateral, in some sense.

Another, more elementary, example is simply the maximum angle of the triangle, which is at least $60^\circ$ (because the maximum is at least the average), with equality exactly for equilateral triangles.

Wikipedia has a list of triangle inequalities which might be a good place to get ideas along these lines.

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As discussed in comments, one nice quantity along these lines is $$ e = \frac{M-m}{M+m} $$ where $M$ is the maximum angle and $m$ is the minimum. Examples:

• $\frac{60-60}{60+60} = 0$ (analogue of circle)

  

• $\frac{90-45}{90+45} = \frac13$ (analogue of ellipse)

  

• $\frac{90-30}{90+30} = \frac12$ (analogue of ellipse)

  

• $\frac{180-0}{180+0} = 1$ (analogue of parabola)

  

• $\frac{120-(-60)}{120+(-60)} = 3$ (analogue of hyperbola)