Geometry: Minimum Length Shape to Enclose 4 Circles (R=1)

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The minimum length of a shape needed to enclose four circles, each with a radius of one, is calculated based on their arrangement. When the circles are arranged so that each is tangent to two others, their centers form a rectangle with sides of length 2, leading to a circumscribing circle with a diameter of 2 + 2√2. The radius of this circle is 1 + √2, resulting in a length of 2π(1 + √2). Alternatively, if considering a shape with rounded edges, the total length around the four circles is 8 + 2π. This analysis provides insights into geometric arrangements and their implications for enclosing shapes.
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what is the minimum length of a shape needed to enclose 4 circles, each with a radius of one?
 
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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
8+ 2π.
 

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