Geometic Series that sums to circle?

[111111]

Does anyone know if there is a way to divide up the area of a circle using similar polygons, with a common ratio? I was just curious if there is a way, or if it has been proven impossible.

For example, I tried inscribing a square inside a circle and making an infinite series of triangles with the remaining area, but the triangles do not have a common ratio.

 

[arildno]

IF we can write the area of the unit disk as the limit of a geometric series, then, with a0 being the area of the largest sub-figure, and k the constant ratio, then we would necessarily have the following equation:

$$\frac{a_{0}}{1-k}=\pi$$

But, since the ratio between rational numbers itself must be rational, it follows that either a0, k, or both must be irrational numbers.

And that sort of deflates the attractiveness of the procedure, don't you agree?

 

[111111]

This probably doesn't make a difference but the area of circle is equal to pi*r^2 where r is the radius of the circle.

But I don't know why that would deflate the attractiveness of the procedure. Why couldn't the ratio be the sqrt(2) or something?

 

[arildno]

I was talking about the "unit disk", with radius equal to 1.

Well, sure it could be 1/square root of two or whatever else, but to construct some nasty irrational number is generally more difficult than a simple rational ratio.