```
2
1----
. |
. |
.|
0 1 2 3
```

Compute the number of points which fall within the unit circle centered at 1.

The quarter-circle that's the part of the unit circle that's in the first quadrant has area $\frac{1}{4}\pi \times {1}^{2}=\frac{1}{4}\pi $, and the unit square has area 1, so about $\frac{1}{4}\pi $ of the random points should be within the quarter-circle.

We compute the ratio of the number of points in the quarter circle and the total number of random points, multiply it by 4, and get $\pi $.

To test whether the point $(x,y)$ is within the unit circle, we test whether ${x}^{2}+{y}^{2}<1$ (since points inside the unit circle all are less than 1 unit away from the origin, so that ${x}^{2}+{y}^{2}<1$.

We can use `random.random()`

to generate random numbers in the interval $[0,1)$. Every time `random.random()`

is called, a new random number is generated.

In [1]:

```
import random
x = random.random()
y = random.random()
print(x, y)
```

`x`

and `y`

:

In [2]:

```
x, y = random.random(), random.random()
print(x, y)
```

Another aside: how to generate a random number between 5 and 12? The following will do the trick:

In [3]:

```
5+7*random.random()
```

Out[3]:

We are now ready to compute an approximation of $\pi $:

In [4]:

```
import random
N = 1000000
count = 0 #count will store the number of random points
#that fell within the unit circle
for i in range(N):
x, y = random.random(), random.random()
if x**2 + y**2 < 1:
count += 1
print(4*count/N)
```