Have you ever heard of a nautical mile? Have you ever wondered what in all the world’s oceans that could be? Keep on reading The Math Dude to learn about the math behind the nautical mile - and its close seafaring relative, the knot.
As a native of the great state of California, I have no choice but to love the sea. Actually, that’s not true at all—a lot of Californians live a hundred or more miles from the ocean (it’s a big state), and are more accustomed to mountains that the big blue…but I digress...
The truth is that I really do happen to love all things related to the ocean: the beach, the breeze, the breaking waves, and (of course) the boats.
Which conveniently brings us to today’s marine-themed questions: What’s a nautical mile? What’s a knot? And for that matter, what do either of these things have to do with math?
Keep reading, because those are exactly the questions we’ll be answering today!.
Review: Arc Length
I don’t want to leave you in too much suspense over this whole nautical mile mystery, so I’m going to cut to the chase: a nautical mile is an arc length. Which explains why we talked about arc lengths last time in preparation for today’s discussion.
If you missed last week’s episode, I highly recommend you check it out before continuing on—it was super awesome, and included fascinating fun facts about arc lengths.
The quick summary from that episode is that the famous formula for the circumference of a circle, C = 2πr, is really a formula for calculating the length of the arc that happens to be a complete circle. And from knowing the fact that 2π radians is the angle around an entire circle, we can surmise that the general formula for the length of an arc (traditionally called s) that has a radius of r and spans an angle θ is given by s = θ • r.
What Is a Nautical Mile?
So what’s the nautical connection? Well, imagine that the Earth is composed of two half-spheres stitched together along a great circle connecting its north and south poles (or any other great circle, such as the equator.) If you pull the two halves apart and focus on just one of the hemispheres, then evenly divide the path around the resulting great circle into 360 degrees of angular distance, and finally, divide each of these degrees into 60 equally spaced minutes of arc, you’ll be all set to understand nautical miles.
A nautical mile is around 1.15 regular miles.
Because, you see, a nautical mile is simply the arc length that you, a ship, or anything else travels along the surface of the Earth when moving an angular distance of 1 arcminute—that’s it!
As it turns out, the circumference of the Earth is around 24,900 miles. And there are 360 degrees x 60 arcminutes/degree = 21,600 arcminutes in a full angular trip around the Earth. Which means that a nautical mile is around 24,900 / 21,600 or 1.15 regular miles—so they’re pretty close.
Nautical miles are convenient for pilots and sea captains, because they often think about distances in terms of angles. When a ship travels from some longitude to another, that distance is really just some number of arcminutes. In other words, it’s really some number of nautical miles.