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It is also related to the densest circle packing of the plane, in which every circle is tangent to six other circles, which fill just over 90% of the area of the plane. The case when the problem is restricted to a square grid was solved in 1989 by Jaigyoung Choe who proved that the optimal figure is an irregular hexagon. [4] [5]
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden.
In geometry, the circumference (from Latin circumferens, meaning "carrying around") is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. [1] More generally, the perimeter is the curve length around any closed figure.
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]
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]
The inner circle is observed to slip with respect to its track. The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2πr, its own circumference with radius r. The inner circle is not ...
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Mrs. Miniver's problem – Problem on areas of intersecting circles; Pivot theorem – Concerns 3 circles through triples of points on the vertices and sides of a triangle; Pizza theorem – Equality of areas of a sliced disk; Squaring the circle – Problem of constructing equal-area shapes