What Is a Nautical Mile?
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.
Jason Marshall, PhD
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What Is a Nautical Mile?
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.
Nautical Miles On the Moon
1 knot is equivalent to 1.15 miles per hour.
If any of you happen to be space travelers, here’s a quick word of warning: if you go to the Moon, or any other object that’s not the exact same size as Earth, the arc length associated with traveling 1 minute of arc will not be an Earth nautical mile.
Why? Because the radius of the moon, or planet, or whatever, will be different…and that means that the arc length in the equation s = θ • r will be different for that 1 arcminute angle. So beware!
If you’re curious, the Moon’s radius is about 3.7 times smaller than the radius of the Earth. Which means that if you were to sail 1 arcminute across the magnificent and completely imaginary lunar seas, you’d have moved only about 0.31 miles, instead of the 1.15 miles you’d have traveled on Earth.
The lesson here is simple: arc lengths depend on the radius of whatever is sweeping out the arc – planets included.
What Is a Knot?
So now you know what a nautical mile is – and if you’re anything like me, you feel a deep sense of relief! I had, after all, lived for several decades occasionally hearing this term and not having the foggiest notion about what it meant…until I quenched that thirst a few weeks ago.
Which brings us to another one of those aquatic-themed terms: the knot. No, not the kind of knot that you tie in a rope (although that certainly does have its own maritime math connections): we’re talking about the kind of knot that sailors and airline pilots declare themselves to be traveling at. What is it?
It turns out that 1 knot is simply the speed you travel when moving at 1 nautical mile per hour. In other words, 1 knot is equivalent to 1.15 miles per hour.
So, for all intents and purposes, you can think of knots and regular old miles per hour as pretty much the same thing. Flying at 500 knots is close to flying at 500 miles per hour. (OK, it’s actually closer to 575 mph…but who’s counting?)
In case you’re also curious as to where this seemingly strange term comes from (as I was), it turns out that ye old-timey sailors used to use a device called a chip log that was essentially a long reel of string with knots tied in it at some specified interval.
When you wanted to find out how fast you were traveling relative to the water, you’d drop the string into the water and start counting knots that passed through your hands. After 30 seconds (or some other predetermined time), you’d take the number of knots, perform the appropriate conversion, and you’d have your speed…in knots.
Wrap Up
Okay, that’s all the maritime math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Old-timey map image courtesy of Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.