Given a circle with area A and many equilateral triangles each with area B, B < A. How many triangles would fit in the circle?

  • $\begingroup$ What did you try? $\endgroup$ – RGS Oct 31 '17 at 20:41
  • $\begingroup$ Provided one such triangle could fit inside (which is not certain), an infinite number of such triangles could be drawn inside the circle. $\endgroup$ – Joffan Oct 31 '17 at 20:42
  • $\begingroup$ @Joffan the area of each triangle is constant. $\endgroup$ – Andrew Tindall Oct 31 '17 at 20:49
  • $\begingroup$ @AndrewTindall Yes, I understood that. What was not stated was anything about overlapping or not. I can draw triangles the same size all day provided the first one fits. $\endgroup$ – Joffan Nov 1 '17 at 0:41

The maximum area $B$ which allows any equilateral triangles of area $B$ to fit inside the circle at all is $\frac{3\sqrt{3}}{8\pi}A$ (the area of an inscribed triangle divided by the area of a triangle). As $B$ gets smaller, the packing gets more efficient, and since equilateral triangles tile the plane perfectly, the number of triangles approaches $\frac{A}{B}$ exactly as $\frac{B}{A} \rightarrow 0$.

Here is a page with a few intermediate values.

Wolfram Alpha will also tell you the minimal side length of the triangles you can pack given a number and a radius, or the number of triangles (and the percent of the circle's area) you can pack given the area of the triangles and the circle. It does it so quickly that it seems there is an algorithm they use, but unfortunately W|A code is proprietary. However, there are a few articles on different methods, like this one.


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