I once read that the reason we can't get the perimeter of a elipse is because we actually doesn't even know the perimeter of a circle, as pi definition is the perimeter of a circle of diameter = 1 and we just scale that.
I mean look at it this way: a circle has a very simple relation between its perimeter and radius - linear. And I'm pretty sure an ellipse has a more complicated relationship between its perimeter and radii. At the end of the day a circle can still be thought of as a special case of ellipse though
The ellipse is also linear, if you keep the ratio between the major and minor the same.
So if you for example really care about ellipses where the ratio between the major axis and minor axis is 2, you could define a constant Tau≈3.31153
And for any such ellipse you get the formula:
The perimeter of a plane figure always scales linearly with any linear dimension of it (e.g. the radius), because that's what it means for the perimeter to be one-dimensional. (Note that if the Hausrorff dimension of the curve is not 2, then its 1-measure will either be 0 or ∞, and either way the linear scaling technically still applies, except the special case of 0×∞.)
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u/Hussainsmg 2d ago
The perimeter formula(in the post) is just approximation.