Area of Ellipse and Volume of Ellipsoid WITHOUT Calculus

Let's utilize an elegant transformation argument to compress ellipse into a circle and ellipsoid into a sphere. We prove the formulas for area of ellipse and volume of ellipsoid.

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Reference: AoPS Volume 2 (7th edition) by Richard Rusczyk and Sandor Lehoczky, which provides the given argument for ellipse in terms of orthogonal projections and implicitly sketches the "limiting rectangle" argument in the pinned comment (see Example 22-1 and Solution to Exercise 22-2, respectively)

[LetsSolveMathProblems]

* At 1:22, the volume should be 4/3 abc * pi, not 4/3 abc. I apologize for the error.

The proof in the video may look informal to a more experienced viewer, and it is indeed informal: the argument in the video cannot be formalized without use of calculus. However, the fundamental idea behind the proof in the video is NOT flawed. To make the proof more rigorous, consider dividing an ellipse into finitely many horizontal rectangles, whose length stretches in the direction of compression. Then, as one and only one dimension of the ellipse is compressed, the compression ratio of ellipse's area can be estimated by the compression ratio of the sum of rectangles' areas, which is equal to the compression factor of length in the direction of compression (this can be easily proven using the area formula for rectangle). Then, note that as we have more and more rectangles (that is, as n, the number of rectangles, approaches infinity), the approximation becomes more and more accurate. Thus, the ratio of compression of ellipse's area is PRECISELY equal to the ratio of compression of the one-dimensional length of any segment in the direction of compression. If you have studied calculus, you know that this process is mathematically valid (AND formal). This is analogous to taking the limit of the Riemann sum to find the area under a curve. For reference, I list AoPS Volume 2 (7th edition) by Richard Rusczyk and Sandor Lehoczky, which provides the video's argument for ellipse in terms of orthogonal projections and implicitly sketches the "limiting rectangle" argument in this comment (see Example 22-1 and Solution to Exercise 22-2, respectively).

One of the reasons I did not mention calculus in the video was to enlarge the size of potential audience. However, there was another important reason: the idea behind the transformation in the video is intuitive enough that even viewers unacquainted with calculus can appreciate it. I did not wish to confuse some viewers with too much formality when the principal reason for the video was to illustrate how an elegant transformation can be intuitively extended to situations involving more complex shapes (other than rectangles). Of course, sometimes the cost of elegance is loss of formality, as is proven here. However, I hope that more skeptical viewers now at least see that the argument in the video, at its foundation, is mathematically valid.

[bugjustbug]

Thank you for explaining this starting with basic geometry terms. I haven't looked at calculus in 6 years and some of the online explanations jump into theories and stuff that I just don't have working knowledge of. I appreciate your understanding that all people deserve access to knowledge even if it isn't the most formal version of the concept. (I am trying to write a scifi book grounded in real science and got stuck trying to imagine an ovoid because I assumed that an ellipse and an oval were the same thing)

[gg.]

Calculus is still hidden behind this approach, as u are using the determinant of the transformation, and the area scale factor of a transformation is the determinant of it by means of calculus essentially

[drsonaligupta]

At 1:22 volume is 4/3abc you forgot pie it should be 4/3pie abc

[architmundada]

Thanks..
Although this may not be considered as 'formal', this really helps to build the reasoning and intuition behind the formula.
Now, when I am using the formula, I am not doing it blindly- and that's what matters to me.

[eamonkerney]

thanks so much this saved my maths project i had been working on for the past few weeks XD

[turtlellamacow]

The scaling factor argument doesn't really bother me because i can imagine dividing the ellipse into thin horizontal rectangles but of course it would take calculus to state it rigorously). What bothers me is: how do we know the resulting shape after scaling is a circle? It can be shown easily by substituting x = (a/b)x' into the equation for an ellipse, but I think it's worth pointing out that it's not obvious (to me) from the geometric definition of an ellipse that scaling it will yield a shape of constant radius :)

[easymathematik]

I would say this is a nice way and method a 16- years-old student could solve this in maths olympics. Why not?

Most of the geometric area and volume formulars one could remember exactly this way.

For an visual explanation in a nutshell more then enough for younger students in middleschool.

In case of emergency one could still deal with all the integration stuff.

Conclusion: nice video. :)

[koenth]

Can you do the surface area of an 'ellipsoid' (x/a)^2+(y/b)^2+(z/c)^2, where a, b and c are different?
I have done them when two of them are the same, but need the general case.

[drewsippel]

I think it would be useful to show that the ratio between the area of a rectangle and the ellipse it inscribes is constant? It's pi/4, I think, but how might one prove that?

[manla]

I never thought about using ratio to derive these.

[hugomoralesg]

If you cut the ellipse by half and look at it from a top view, is the surface area = pi*a*b ??

[jakobkrausz]

Why not get the average of both a and b (meaning both radiuses r1 and r2 formed by inequal diameters) and multiply by pi?

[allanlago]

Can this same argument be used to find the perimeter/surface area of an ellipse/ellipsoid too?

[reganreynolds]

is there a way to find another radius if I've only been given the diameter and c?

[Czeckie]

not convinced, you have to prove that stretching in one direction is affecting the area/volume linearly.

[rohitg]

Hmm, I get that for a rectangle, the change in area is proportional to the side you are changing because I already know that the formula is ab. However, when doing so for an ellipse, aren't we assuming that the formula for the area is kab, where k is some constant?

[redambersoul]

Reading to the comments below I do find a good argument "against" that video (unfortunately):

i.e. an average (mathematical) educated person will be impressed and leave the discussion at that point and miss all the real mathematics behind. Adding a part at the end, that this thought holds only true for area and volume but not for circumference and not for surface would put it in the right perspective. Discussing the limitation of such thoughts or easy and nice "tricks" is something I love so at other channels like 3Blue 1Brown and Mathologer.

[iamtrash]

Seems like calculus to me though. Since essentially we have to treat a sum of areas of small squares whose area tends to 0 for it to work

[peasant]

is this correct proof though? I mean is it mathematically accepteD?