We all know that in our math and physics classes 3.14 is a sufficient substitute for pi. But how much error does this method produce? What does NASA use for their calculations? According to the chief engineer for NASA’s Dawn Mission Marc Rayman, pi to the fifteenth decimal place is sufficient enough for all calculations with error that is very insignificant. For example, if we use the distance from the Earth to the most distant spacecraft, the Voyager, as the radius of a circle, the calculated circumference using fifteen decimals of pi gives an error that is less than the size of your little finger. The most distant spacecraft is the Voyager 1, about 12.5 billion miles away. If use that as the radius of our circle, and pi rounded to the fifteenth decimal, 3.141592653589793, the value of the circumference come out to be 78 miles. It turns out that this value is off by 1.5 inches, which has a percent error of 0.00003035128%, extremely insignificant.
If the error for such a big value is this small, calculations done on earth with this value of pi give numbers that are off by only a molecule. For example, if we use the diameter of earth as 7,926 miles, it gives a circumference of 24,900 miles. From this, we can see that by not using more than 15 digits of pi the error would be 10,000 times thinner than a strand of hair!
If we look at the largest size, which is the visible universe, how many digits of pi would be needed to calculate the circumference of a circle with the radius of the universe so the error is as small as the diameter of a hydrogen atom? The radius of the universe is about 46 billion light years. In fact, to calculate this circumference to a higher degree of accuracy, you would need about 40 decimals of pi. However, if you compare the huge vastness of the universe to the amount of decimal places of pi needed to calculate it, we do not indeed need to use many digits of pi to cover such a huge range.
Fun Fact: 22/7 is used as a better approximation for pi than 3.14 (even though the decimals places don’t exactly perfectly match up), so if you want a better estimate of pi without inserting all the numbers in your calculator, you can just input this value!
