How come $\pi$ is usually approximated as 3.14 or 22/7? I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857....  Guess what?  $\pi$ can also be approximated as 355/113, which is equal to 3.1415929203539823008849557....  There are 112 numbers after the decimal, which then start repeating.  Anyway, let's cut to the chase.  Why is $\pi$ usually approximated as 3.14 or 22/7?  Maybe they're close to the actual result?  Anything else about it?  I do know that these can be used to find the circumference or area of a circle.
 A: Say you're trying to approximate a number by rational numbers $p/q$. Usually, the bigger $q$ is, the better your chance of approximating the number closely. On the other hand, the smaller $q$ is, the simpler the approximation.
In the case of $\pi$, if you want to have a better approximation than $22/7$, you have to go all the way up to $q = 57$. (See this.) So $22/7$ is a remarkably accurate approximation, considering how low its denominator is.
A: 3.14 is two decimal places. That's the only justification I can give. When doing hand calculations (or sliderule calculations), carrying out more than three significant digits is cumbersome. Remember this: with $N$ digits in a multiplication, you have $N^2$ digits in the final answer before you can round again. In the world of engineering, often you can't (afford to) manufacture something with insane tolerances. Basically: precision=money. So carrying out more digits might not make sense. For another example, imagine you are estimating the area of grass in a circular front yard (landscape designers, please don't troll. ;)) in order to put down fertilizer. Going from $3.14\rightarrow3.1416$
 is two more digits, but are you seriously going to measure pounds of fertilizer down to the $\frac{1}{10,000}$ of a pound? That's like a gallon of water $\pm<\frac{1}{10}$ of a teaspoon.
22/7 is easier to remember. It's $\frac{21+1}{7}$.
Basically, this goes back to the days of slide-rules and hand calculations. As @user_of_math pointed out, we don't need to truncate today with computers as fast as they are, but there is now the trap of artificially high precision.
A: 355/113 is actually easy to remember (113355, written bottom to top), and is the best approximation with a denominator less than 5 digits, accurate to six decimal digits. I always wonder why we don't learn that value, being both remarkably accurate and easy to memorize. 
