A square and a circle with the same area

Short Version

• The length of the diagonal of the square is larger than the diameter of the circle.

• The circumference of the circle is less than the perimeter of the square.

• The circle is too large to be inscribed in the square.

• The radius of the circle is larger than half the length of the square's sides.

• It is the ratio of the squares of the side length of the square to the the radius of the circle that is $\pi:1$

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Details

If $a$ is the side of the square and $r$ is radius of the circle, equal area translates into: $a^2=\pi r^2$ , or $a=r\sqrt{\pi}$ and $r=\dfrac{a}{\sqrt\pi}$

• The length of the diagonal of the square $=a\sqrt2=r\sqrt{2\pi}\approx2.507r>2r=$ diameter of the circle.

• The circumference of the circle $=2\pi r=2\pi\times\dfrac{a}{\sqrt\pi}=2a\sqrt{\pi}\approx3.545a<4a=$ perimeter of the square.

• The radius of the circle $r=\dfrac{a}{\sqrt\pi}\approx0.564a>0.5a$ , which is what it would have to be to be inscribed in the square.

• The radius of the circle $r=\dfrac{a}{\sqrt\pi}\approx0.564a>0.5a=$ half the length of the square's sides.

• The ratio of the side length of the square to the radius of the circle is $\dfrac{a}{r}=\sqrt{\pi}:1$ not $\pi:1$

Only $2$ is true. I will provide more explanation later on.