# How many triangles fit in a circle

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

• What did you try? – RGS Oct 31 '17 at 20:41
• Provided one such triangle could fit inside (which is not certain), an infinite number of such triangles could be drawn inside the circle. – Joffan Oct 31 '17 at 20:42
• @Joffan the area of each triangle is constant. – Andrew Tindall Oct 31 '17 at 20:49
• @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. – 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 $\frac{B}{A} \rightarrow 0$.