Divide a Circle into Five Equal Areas

CHALLENGE:

Find many different ways to divide a circle into 5 equal areas in ways that the perimeter of none of the areas goes through the center of the circle.For example, on the right, the concentric bands mare off five equal areas.Here are some others . . .

For each pattern you identify, determine whether it can be created with ruler and compass constructions (or GSP). For instance, the ones with a square in the center that is one-fifth of the area of the circle can not be "constructed" -- the quadrature of a circle (constructing a square of the same area) is one of the classic problems of antiquity that has been proved to be impossible with straightedge and compass. So in such examples, some measuring device would have to be used.

On the other hand, the example with the concentric rings at the upper right, can be constucted. This figure shows one construction of the radii for the respective circles, beginning with a circle of radius r and then construction radii of the other circles.

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