Archimedes, about 2300 years ago, showed that the rational number 22 7 is greater than π , so it is a potential candidate. Let’s see if it’s less than √ 10 . We can do that by seeing if it’s square is less than 10 .
( 22 7 ) 2 = 22 2 7 2 = 484 49 < 490 49 = 10 .
So it looks like the very common rational approximation of π is in fact a rational number between π and √ 10 .
By the way, it appears that the second most common rational approximation of π , namely 355 113 , is also greater than π , so forms a second answer to this question, with π < 355 113 < 22 7 < √ 10 .
