A square and two rectangles, connected by two circular arcs, form a square. What is the proportion of the orange to the green area?
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Solution
The green and orange areas have the same size.
![](https://mirangu.com/wp-content/uploads/2020/11/WindshieldWipersSolution.jpg)
Setting the side of the orange square to 1, it is easily seen from the Pythagorean Theorem that the circular arcs have radius √5/2. Now this makes the large square have side (√5+1)/2, which is the Golden Ratio φ.
The green rectangle has area φ2-φ, which is known to be 1, exactly that of the orange square.