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If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged. An elongated shape can be made more round while keeping its perimeter fixed and increasing its area. The classical isoperimetric problem dates back to antiquity. [2]
Roundness = Perimeter 2 / 4 π × Area . This ratio will be 1 for a circle and greater than 1 for non-circular shapes. Another definition is the inverse of that: Roundness = 4 π × Area / Perimeter 2 , which is 1 for a perfect circle and goes down as far as 0 for highly non-circular shapes.
The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. The solution is intuitive; it is the circle. In particular, this can be used to explain why drops of fat on a broth surface are circular. This problem may seem simple, but its mathematical proof requires some sophisticated theorems.
Alternatively, the shape's area could be compared to that of its bounding circle, [1] [2] its convex hull, [1] [3] or its minimum bounding box. [3] Similarly, a comparison can be made between the perimeter of the shape and that of its convex hull, [3] its bounding circle, [1] or a circle having the same area. [1]
United States Ryder Cup players will begin receiving a $200,000 "stipend" for playing, starting next year. (AP Photo/Alessandra Tarantino, file) (ASSOCIATED PRESS)