9846

Pi-Like Ratios for Circular Lakes

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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