A calculation that goes awfully wrong if we let $\pi=22/7$

Question

Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation in which if we use $\pi=22/7$, some noticeable error which makes quite a significant difference, appears. Kind of calculative contradiction.

Thanks!

Answer 1

Consider calculating the position of a satellite in a circular orbit about the earth. After a few hundred rotations, the satellite will be noticeably in the wrong place.

Answer 2

$$\int_0^1 dx \frac{x^4 (1-x)^4}{1+x^2} = \frac{22}{7}-\pi$$

That integral is clearly not zero.

Answer 3

Compute the third decimal digit of $\pi$.

Perhaps even better, because it is as much wrong as it can be:

Compute the ninth binary digit (in the fractional part) of $\pi$

Answer 4

What quadrant is $1\ 000\ 000$ radians in? The answer you get to that question is wrong if you use $22/7$ as an approximation to $\pi$.