Deriving the value of $\pi$ from a dart board

Question

I saw this on a website and it was pretty interesting:

The circle inscribed in the square has a radius of $1$ and the square has a side length of $2$. This means that the area of the circle is:

$$\pi \times r^{2} = \pi \times (1^2) = \pi$$

Moreover, the area of the square is:

$$(2^2) = 4$$

Then, we can randomly select points within the square and get a good approximation of pi by doing this thousands to millions of times, maybe by a computer sequence. Once we've repeated this process enough, we can plug our values into this equation:

$$ \frac{\pi}{4} = \frac{\text{Number of points within the circle}}{\text{Total number of points}}$$

Is this a viable approach to approximate the value of $\pi$?

Answer 1

Interesting. I just wrote a simple Python program with $10,000,000$ 'dart throws'. Here are the values of the estimate from five runs

$$3.141466, 3.142664, 3.1420584, 3.1422768, 3.1417728$$

So qualitatively, I'd say this is not a great method. By contrast, there are a number of series that can arrive at 4 significant figures of $\pi$ with far less calculation.

Answer 2

Yes, this is a valid approach to compute $\pi$ provided the points are drawn randomly using the uniform distribution on the square.