You can do a very quick calculation of get some very rough bounds on pi.
We know that if a circle has radius 1 then its area is pi. Now we can completely surround that circle with a square with side length 2 (draw a picture, it will be clear why), and the area of this square is 4. hence pi is less than 4.
Alternatively we can draw a square with diagonal length 2 and fit it inside the circle. A slightly more complex calculation gives the area of that square as 2. So immedaitely we know that pi must be between 2 and 4.
You can use more complex shapes like pentagons, hexagons etc and get even tighter bounds.
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