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I am looking into elliptical wings at various aspect ratios and my intuition tells me that an elliptical wing with aspect ratio of one is a circle, meaning the chord and the span will be equal. But, when I lay down the math, this is not true. For example, $$ AR=1=\frac{span^2}{area}=\frac{span^2}{\pi (span/2)(chord/2)} $$ $$ {\pi (span/2)(chord/2)}={span^2} $$ $$ chord=\frac{4*span}{\pi} $$

This tells that if the span is 100, then the chord will be 127 approximately for an aspect ratio of one, which is not a circle. Is my math correct, is an elliptical wing with an aspect ratio of one not a circle?

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is an elliptical wing with an aspect ratio of one not a circle?

Aspect ratio is wingspan divided by average wing chord.

Let W = wingspan.

A perfectly square wing would have an aspect ratio of 1, because the wing chord at any point = W, and thus the average wing chord = W.

Take a perfectly square wing and draw a circular wing inside of it, with the same wingspan, and the same root chord, as the square wing. Clearly the average wing chord of the circular wing is less than the average wing chord of the square wing, so the aspect ratio of the circular wing must be greater than 1.

If we took the circular wing and "compressed" or "compacted" it into an ellipse (with wing root larger than wing span) by pushing on the wing tips, at some point we'd eventually achieve an aspect ratio of 1. But that wing would clearly no longer be a circle.

So no, an elliptical wing with an aspect ratio of 1 is not a circle.

What is the aspect ratio of a circular wing?

Aspect ratio = span squared / area.

Let the wingspan be equal to 1 arbitrary unit, which is twice the wing radius. In these same units, the wing area is equal to pi times ((.5) squared) which works out to .785. This gives an aspect ratio of 1 / .785, which is 1.274.

PS the math in the original question is correct, as long as we keep in mind that the term "chord" is being used there in a way that actually means "root chord".

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  • $\begingroup$ Since π is used for area but not span, the aspect ratio of a circle is 4/π. Your result is good, but presenting it as a function of π might be more revealing. Of course, the question already presents it that way. $\endgroup$
    – Peter Kämpf
    Commented Feb 1, 2022 at 14:33
  • $\begingroup$ I only answer when nobody steps forward or the existing answers are poor. So I saw no need to add my own here. $\endgroup$
    – Peter Kämpf
    Commented Feb 1, 2022 at 15:26
  • $\begingroup$ Thank you very much for your explanation, the square wing example was helpful! $\endgroup$
    – August
    Commented Feb 1, 2022 at 16:10

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