If you really want to use this method to calculate pi, here's how to actually go about it. What you need is a hundred yards or so of string, four stakes, a stick and something that's a reasonable approximation to a right-angle (perhaps a piece of a cardboard box salvaged from the apocalypse). If you're really stuck for a right angle you can construct one with three stakes and a piece of string by putting two stakes in the ground and using the string to mark a straight line between them, then tying one end of the string to one of the stakes and tying the third stake to the string, so that length of string between them is a bit over half the distance between the stakes in the ground. Mark out a circle using this. Then mark out a second circle with the other stake in the ground as the centre. These two circles will intersect at two places - use the string to mark a straight line between them. The two straight lines you have marked will be at right angles.
Now put two stakes in the ground, about 20 yards apart. Stretch string between them. Put your right-angled thing with one side against the string and the right-angle corner at one of the stakes. Measure another piece of string to be the same length as the piece stretched between the two stakes. Tie it to a third stake and stretch it out so that it runs along the other side of the right-angled thing. You've now marked out two sides of a square with string. Repeat to form the other two sides.
Take your stick and break it down to about a foot long. Use it to mark out on the ground equally-spaced marks along each side of the square. Get two people to hold each end of a fifth piece of string across the square so that you can mark straight lines on the ground, dividing the square into a grid.
Cut your fifth piece of string to be the same length as one side of the square. Tie one end to one of the stakes. Now use the other end to mark out an arc from one corner of the square to the opposite corner.
Count the number of squares that are inside the arc and the total number of squares. Take the ratio of these two numbers and multiply it by 4. Here is your approximation to pi.
This method has many advantages over the one proposed: With the dimensions given above, it gives a considerably better answer, correct to four significant figures (3.141). It is easy to scale for better accuracy - make the square 100 yards and the stick four inches and you get six correct digits (3.141590123). You don't need to correct for uneven shot pattern. And, crucially I'd say in an apocalypse, you don't need a shotgun or ammunition and, if you do happen to have them, you can use them for useful things like fending off the zombies or hunting.