what is the minimum length of a shape needed to enclose 4 circles, each with a radius of one?
How are the circles spaced? Can they overlap at all?
Assuming you have all four circles abutting, so that each is tangent to two others, their centers form a rectangle with sides of length 2. The distance between the centers of diagonally opposite circles is 2√(2). The distance along that diagonal, extended, from the center of such a circle to the outer edge of its circle is, of course, 1 so the distance along that diagonal, from the outer edge of one circle to the outer edge of the other is 2+ 2&radic(2) and that is the diameter of the circle circumscribing all 4. It's radius is 1+ &radic(2). That would have length 2π(1+ √(2)).
If you are only concerned about a "shape" that includes all four circles, you have a square with "rounded eges". The straight portion of each has length 2 (and there are 4 of those). The rounded portion is a quarter of each circle: length (1/4)(2π)= π/2 and there are four of those: the total length around the four circles is
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