Square area equal to difference between the circumscribed circle and the inscribed circle

Question

There is no such square whose area is equal to the difference between the areas of its circumscribed and inscribed circles. Am I right?

Imagine a square with length $L$.
This square has an area of $L^2$.
Its inscribed circle has radius $\frac{L}{2}$.
Hence, it has an area of $π \frac{L^2}{4}$.
By Pythagoras' theorem, the circumscribed circle has a radius of $\sqrt 2L$.
So it has an area of $2πL^2$.

A.T.Q. :
\begin{align} \text{Area of circumscribed circle - Area of inscribed circle} &= \text{Area of a square} \\ 2πL^2 - π\frac{L^2}{4} & \neq L^2 \end{align}

Does this complete my proof?

Circumscribed and inscribed circles of what, exactly? Any convex polygon? — , Dec 06 '17 at 14:40
"If any one wants to know my proof just request it." - That's not how [tag:proof-verification] works. — user202729, Dec 06 '17 at 14:42
Also, probably you meant "constructible in finitely many steps with straightedge and compass". — user202729, Dec 06 '17 at 14:43
Typically, posts tagged "proof-verification" include a proof, with the question, "Is my proof correct?" Since I don't see your proof, I'm not sure what it is that you want us to do... — Xander Henderson, Dec 06 '17 at 14:43
The polygon I mean is the square. And I am quite unsure whether my profile is right. Anyway I am adding my proof. — For the love of maths, Dec 06 '17 at 14:44
@user202729 This isn't about constructability, but the area of the annulus given by the circumscribed and inscribed circles of a square, compared to the area of the square itself. — Arthur, Dec 06 '17 at 14:44

score 1 · Accepted Answer · answered Dec 13 '17 at 09:51

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It is right, I suppose? π is not equal to 4.

answered Dec 13 '17 at 09:51

For the love of maths

2,368

See your revised post. In it you provided a counter-example, which, indeed, proves your assertion. For rigorousness, you may want to remark that every square is completely defined by the length of its sides. – Mussé Redi Dec 28 '17 at 16:35
You're very welcome. – Mussé Redi Dec 28 '17 at 16:36
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In fact I expanded on this idea: https://math.stackexchange.com/questions/2579805/no-polygon-has-the-same-area-as-the-difference-between-its-inscribed-and-circums/2580922#2580922 – For the love of maths Dec 28 '17 at 16:37

Square area equal to difference between the circumscribed circle and the inscribed circle

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