I How can I derive pi?

Derive what?
You first need a definition of pi. Everything else (what exactly do you want? Its numerical value?) has to start from there.

 

There are various ways to calculate approximate values. That is the result of 10 seconds google search...

 

Take a tiled floor and scatter a handful of nails or satay sticks, mix with lots of patience, and you can experimentally determine a value for Pi by a procedure known as Buffon's Needle experiment. http://www.math.leidenuniv.nl/~hfinkeln/seminarium/stelling_van_Buffon.pdf

Don't expect many places of accuracy at first, but to improve on this, rinse and repeat many times.

And did I mention that you'll need lots of patience?

 

Another demonstration uses what is known as a Monte Carlo technique. Draw a square of some convenient size, and inside the square draw the largest possible circle.

Next, scatter lots of dots at random all over the square; keep a count and call this number S dots. Count how many of those random dots fall inside the circle, call this number C. Then the ratio C/S gives us the ratio of the areas of the two figures, viz., Pi/4.

The more dots you scatter, the more accurate the resultant estimate for Pi. Repeat this multiple times to improve the result even further.

I like this method, because after performing it 7 or 8 times you can apply a law of statistics to produce an estimate of Pi to much better precision than you would expect.

 

To expect to get 3.14 correct (<1/100 error on 4-pi), you need about 10,000 points. 3.141 or 3.142? 1 million. A computer can easily do that, but it doesn't sound like fun to do it manually.

 

Which provides an incentive to improve the procedure by reducing the labor involved! You don't need the full common central area, so you can eliminate the circle's inscribed square from the picture. And you don't need 4 areas, all identical---just one will do. This leaves us with a 3-sided figure containing a circle segment: the competing areas to be covered with dots now being roughly equal.

I'm not sure how to prove it mathematically, but I have a notion that when there is not a big difference in the two competing areas then Monte Carlo gives us better accuracy with fewer random dots. I think I decided that by taking the ratio C/S for this modified figure, and doubling it, we have the fractional portion of Pi, i.e., our estimate to ⋅14159

 

I can help, because we have a binomial distribution with variance ##N p q## whereas the central value is Np or Nq. You gain much more from removing areas, however - going to a triangle makes the final formula ##\pi = 2 + 2 \frac{inside}{total}##, so the relative uncertainty now applies to a smaller value (~1.14 instead of 3.14). You could now start to cut away a triangle from the outer part as well, going to ##\pi = 2 + 1.3 \frac{inside}{total}## and so on. Effectively you are approximating pi with the areas itself then, the closer you get with one of the areas the better.

 

Even with 500 points, getting 3.14 is more luck than a good estimate. 3.1 or 3.2 should be very frequent, but still not guaranteed.

 

Sure, you can use a ruler. Other methods are just more accurate.