Square and Circle Observation

Squares and circles interest me, or more precisely the relationship between the square and the circle interests me. Let’s not go down the rabbit hole of ‘squaring the circle.’ This is a problem as ancient as geometry itself, but thanks to Ferdinand von Lindemann π was proven to be a transcendental (non-algebraic and therefore non-constructible) number in 1882, definitively ending the argument – the circle cannot be squared.

No, what piques my interest is that there is a relationship between the circle and the square. The equations of the areas of the square and the circle bear this out: n2 and πr2 both possess the same component when n=r.

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We can see that when the n value of the side of the square is equal to that of the r or radius of the circle then there is a clear relationship; minus the multiplication by π in the case of the circle. We also note that 4 of the same squares will cover the entire area of the circle with each outer line of the squares being tangent to the circle at 4 points.

From simple observation we can also see that the quarter of the circle inside the square takes up close to 23 of of the area of the square, leaving about 13 out of the circle. Given that the r=5, we see that the area of the circle is 25π. The square would then have an area of 25. 25π is roughly 78.54, leading us to ask how close the area of 4 squares minus approximately 13 would be:

4(52) = 100
100(23) = 66.666
That’s about 84.87% of 25π

That’s close-ish. Nowhere near as close as mathematically precise, but it is a constant ratio. It remains 84.87% no matter the size of the radius and the square of it. We will also find that 2.363(4n2) brings us so much closer to the area of the circle when n=r. Of course this is because 2.363(4) = 3.1466…, not too far off of the value of π (3.141…). It doesn’t solve much, but it is interesting.

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