Let's say circle $\omega_1$ has a diameter $X$. Let $X>Y$; $Y\in \mathbf{R}^{+}$. How many circles with diameter $Y$ will fit inside $\omega_1$?
Is there a formula for this?
[Math] The number of circles that will fit inside the area of larger circle
circlesgeometrypacking-problem
Related Solutions
In the image above are sequential steps of how I did it (left image is the earlier steps and the right the later steps). I'll only explain the figure constructed in the center. After having divided the disk into three (left figure) by the three radius, construct an arbitrary chord S perpendicular to AB. Construct circle centered at T tangent to the given circle at U. Construct a line passing U and the point at which the perpendicular and the the circle centered at T intersects. The line should intersect at diameter AB which is the tangent. The rest is easier to follow from the figure.
I was looking for the answer for this problem in the net but to no avail. After much trial and error, got this construction. Not 100% sure though, kinda new to this non-coordinate based geometry.
If your goal is to make a program, then considering you already have that done, it seems like the easiest strategy would just be to: 1. Add on more rings until you know that every circle in a new ring will be outside the main circle. 2. Iterate through the small circles, removing all of the circles who's centers are further than R-r from the main circle, where R and r are the radii of the large and small circles respectively.
That however assumes that you can't fit more circles in by translating your entire set of small circles to the side a bit more. If you want to make sure you have the maximum, you might have to do some more fudging.
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Best Answer
$${{\pi {X^2}} \over {\pi {Y^2}}}$$
=$${{{X^2}} \over {{Y^2}}}$$