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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