If we use the single digit version in which π = 3, what would happen to our height calculation? The answer: nothing.
Remember that the basic trig functions (sine, cosine, tangent) are just ratios of sides of right triangles. If you have a triangle with an angle of 34 degrees, then the ratio of the opposite side to the adjacent side is always 0.6745. So if you change the value of π nothing happens. It’s still a right triangle and still has the same ratio of sides.
But how do we find these values of sine, cosine, and tangent for different angles? The oldest way is to just look them up in a trig table. These are just printed lists with angles and their corresponding sine, cosine, and tangent values. Your pocket calculator does something similar—usually a combination of a look-up table and an approximation of a type to get you that value of tangent (34 degrees). But it doesn’t depend on the value of π.
How Many Digits of Pi Does NASA Use?
Let’s see if the number of digits matters when you’re calculating something vast, like a distance in space. For most calculations, NASA uses 15 digits: 3.141592653589793. Is that enough? Well, here is the full answer from NASA’s Jet Propulsion Laboratory, but I will give you the short answer.
In the NASA answer, they describe the digits of pi with an example using the Voyager 1 spacecraft at a distance of 12.5 billion miles away from the Earth. (Actually, that answer was created in 2015, and Voyager is now more like 14.5 billion miles away.) But let’s think of that as Voyager’s distance from the sun—it’s pretty close to the same thing.
So we can imagine this enormous distance as the radius of a huge circle centered on the sun, as if Voyager was in circular orbit around the sun. We can calculate the circumference of this circle by using 2πR. (I’ll use R = 14.5 billion miles.) Using 15 digits of pi gives a circumference of something like 91 billion miles, which is very long. If you use more digits of pi—like, say, 21 digits—the circumference would actually be longer.
But here’s the important part: Even with 6 more digits, you only get a circumference that’s 5.95 inches longer. Could you imagine measuring 91 billion miles and only being off by less than half a foot? That’s super accurate. So there’s not much point in calculating beyond the fifteenth digit. The returns really diminish beyond that point.
But what about just using 1 digit? If you used a value of 3 for π, that would make a circumference that is 9.1 billion miles shorter. Yes, I think that makes a difference.
So, just to be clear—in this case, 1 digit is not enough, and 15 digits is good enough for everything you can imagine. It’s even good enough for NASA.