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Note that this open investigation can be taken to many levels of complexity.
A large circle of unit radius is constructed. From this initial circle, the following diagram is constructed using only straight edge and compasses :
All circles touch or intersect at tangents only. The initial circle has an area of $\pi$ units squared - this is an irrational area.
Hidden in the image is at least one region with a rational area. Can you find one?
This image could be extended in many ways. How many regions of rational area could you construct using only straight edge and compasses? What interesting images can you construct? What questions do these generate in your mind?
For more investigations see our Stage 5 pages.