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Everybody knows ... for a circle. But for a circular lake on the Earth, the ratios are not constant. In the case of a lake, let ... be the length of the shoreline, ... its area, and ... the distance along the surface from the center to the shore. Let the radius of the sphere be ... . The radius of the Earth is 3963.192 miles. A circular lake with lake radius ... miles has ... . Perhaps whoever was thinking of defining ... to be 3 had this lake in mind. For a circular lake on a sphere with radius ... and lake radius ... , ... , and ... . The ratios ... ... and ... are candidates for " ... " for a circular lake. All four of their limits are ... as ... or ... (as the lake flattens and becomes more like a disk). The ... slider ranges from 1 to 3963.192, from the radius of a unit sphere to the radius of the Earth in miles. Set ... and move ... to be between 0 and ... . The circular lake ranges from a point to all of the sphere except a point. When ... , the circular lake is the upper hemisphere and ... When ... , both ratios are a little less than 3. Move ... , the ... imitator, to determine values of ... ... and ... for which ... and ... , for ... from 1.4 to 3. Then set r equal to ... or ... to see what these lakes look like.

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      EUN,LOM,LRE4,work-cmr-id:397874,http://demonstrations.wolfram.com:http://demonstrations.wolfram.com/PiLikeRatiosForCircularLakes/,ilox,learning resource exchange,LRE metadata application profile,LRE

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