Circular Harmony
The figure above is composed of circular arcs with
.
Is it true that all of these colored regions have the same area?
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1 solution
From symmetry, it is obvious that the blue and the red areas are equal.The area of the green shape is the difference between a circle of 2/3 diameter and a circle of 1/3 diameter. Area is proportional to the square of the diameter, so the green shape is (4/9 - 1/9) of the total area. That means the green shape is 1/3 of the total area, and both the red and blue shapes are 1/2 of the other 2/3, or 1/3 each. Hence, they are all equal.