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I was asked this question on another forum and was interested in it... It's somewhat related to what I have been doing lately so I gave it a (few) tries, but I never really worked it out...

Consider a circle with a radius of 32 units. We want to divide the area of the circle into 9 areas that have, if possible, exactly the same area. See the following image:

The red lines are the 'dividing lines', spaced by a distance [itex]d[/itex] (in the x as well as the y direction).

The areas 1 (blue) and 2 (green) and the area 3 (red) are marked with the colors. Note that there are four areas 1 and four areas 2, they should be equal in area.

The question is how to find the distance [itex]d[/itex] that will yield the optimal result (if possible, that all areas are equal).

The first thing I thought about (but which doesn't seem to be working, see later) is simply to do the following:

We know the area of the complete circle: [tex]A_{tot} = \pi 32^2[/tex]

Therefore, if the 9 areas are to be divided in equal areas, the area of one the subareas will be: [tex]A_{sub} = \frac{ \pi 32^2}{9}[/tex]

We also know the area [itex]A_3[/itex] since it's just a square: [tex]A_3 = d^2[/tex]

Therefore: [tex]d = \sqrt{ \frac{ \pi 32^2}{9}}[/tex].

I tried to graph it and it seemed alright to the eye, but I wanted to be sure, so I went on...

The following way I could think of was to calculate the subareas seperately using integrals and then looking for a [itex]d[/itex] that would minimize their deviation.

I came up with the following area's; [itex]A_1[/itex] is calculated from the top-right area1 and [itex]A_2[/itex] is calculated from the rightmost area2.

[tex]A_1 = \int_\frac{d}{2}^b \left( \sqrt{ 1024-x^2} - \frac{d}{2} \right) \, dx[/tex]

[tex]A_2 = 2 \times \left( \int_b^{32} \sqrt{1024-x^2} \, dx \right) + d \sqrt{1024-\frac{d^2}{4}}[/tex]

[tex]A_3 = d^2[/tex]

where the limit b is the intersection of the circle with y = d/2:

[tex]b = \sqrt{1024-\frac{d^2}{4}}[/tex]

When I now plugged in the value for [itex]d[/itex] I found above I don't get the same result, I get a different result for each area...

So, I thought, maybe my simple solution above wasn't right.

But now I have found three areas each as a function of d. I should be able to minimize the deviation between the areas for one value of d, right? I can't see any way how to do that though... Maybe taking the absolute value of the deviation (A_1 - A_2 for example) and using solving it's derivative for 0? Even then I only minimized A_1 - A_2 and had nothing to do with A_3...

Where have I gone wrong:

1) Assuming there is a solution where all areas are equal;

2) Assuming this solution was simply to divide the total area by 9 and equaling this to d^2;

3) Calculating the areas using integrals?

I can't see any other mistakes I may have made, so I assume it must be one of the three...

Could anyone help me out here?