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
. 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
are candidates for "
" for a circular lake. All four of their limits are
(as the lake flattens and becomes more like a disk).
slider ranges from 1 to 3963.192, from the radius of a unit sphere to the radius of the Earth in miles.
to be between 0 and
. The circular lake ranges from a point to all of the sphere except a point.
, the circular lake is the upper hemisphere and
, both ratios are a little less than 3.
imitator, to determine values of
from 1.4 to 3. Then set r equal to
to see what these lakes look like.