The area of the circle passing through the points 5 , ± 2 , 1 , 2 is

Plotting the diagram of given points we get,

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Now $AB=\sqrt{\left(4^2+0^2\right)}=4$

And $AC=\sqrt{\left(4^2+4^2\right)}=4\sqrt{2}$

And $BC=\sqrt{\left(0^2+4^2\right)}=4$

$\therefore A{B}^2+B{C}^2=A{C}^2$ {as $16+16=32$}

$\therefore$ It is right angled triangle.

$\therefore$ Diameter $=\left(AC\right)$ or hypotenuse

So, length of Diameter $=4\sqrt{2}$

$\therefore$ Radius $=2\sqrt{2}$

Now area will be ${\mathrm{\pi r}}^2=\mathrm{\pi }{\left(2\sqrt{2}\right)}^2=8\mathrm{\pi }$