Circle packing in an equilateral triangle

Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack Template:Mvar unit circles into the smallest possible equilateral triangle. Optimal solutions are known for Template:Math and for any triangular number of circles, and conjectures are available for Template:Math.

A conjecture of Paul Erdős and Norman Oler states that, if Template:Mvar is a triangular number, then the optimal packings of Template:Math and of Template:Mvar circles have the same side length: that is, according to the conjecture, an optimal packing for Template:Math circles can be found by removing any single circle from the optimal hexagonal packing of Template:Mvar circles. This conjecture is now known to be true for Template:Math.

Minimum solutions for the side length of the triangle:

Number of circles	Triangle number	Length	Area	Figure
1	Yes	$2\sqrt{3}$ = 3.464...	5.196...
2		$2+2\sqrt{3}$ = 5.464...	12.928...
3	Yes	$2+2\sqrt{3}$ = 5.464...	12.928...
4		$4\sqrt{3}$ = 6.928...	20.784...
5		$4+2\sqrt{3}$ = 7.464...	24.124...
6	Yes	$4+2\sqrt{3}$ = 7.464...	24.124...
7		$2+4\sqrt{3}$ = 8.928...	34.516...
8		$2+2\sqrt{3}+\frac{2}{3}\sqrt{33}$ = 9.293...	37.401...
9		$6+2\sqrt{3}$ = 9.464...	38.784...
10	Yes	$6+2\sqrt{3}$ = 9.464...	38.784...
11		$4+2\sqrt{3}+\frac{4}{3}\sqrt{6}$ = 10.730...	49.854...
12		$4+4\sqrt{3}$ = 10.928...	51.712...
13		$4+\frac{10}{3}\sqrt{3}+\frac{2}{3}\sqrt{6}$ = 11.406...	56.338...
14		$8+2\sqrt{3}$ = 11.464...	56.908...
15	Yes	$8+2\sqrt{3}$ = 11.464...	56.908...

A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.

See also

• Circle packing in an isosceles right triangle
• Malfatti circles, three circles of possibly unequal sizes packed into a triangle

References

Template:Elementary-geometry-stub