How to Calculate Arc Length
Do you know what an arc is? Do you know that an arc has a length? And do you know how to calculate that arc length – or why you might want to? Keep on reading The Math Dude to find out!
Jason Marshall, PhD
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How to Calculate Arc Length
I’ve been doing lots of DIY projects fixing up La Casa de Math Dude recently, which also means I’ve been listening to a bunch of audiobooks to help pass the time between mind-numbing strokes of the paint brush. During a marathon listening session this week, the term “nautical mile” found its way into my ears – which got me wondering what it means.
Is it really a mile, like the mile we commonly know and love? What’s so nautical about it? Do you have to be a sea captain to care? As I began my investigation, I quickly learned that the nautical mile has a lovely and simple mathematical (combined with a dash of astronomical) origin—and thus is something we should definitely talk about.
But in order to really understand the nautical mile, we first need to understand an idea in geometry called “arc length.” And in particular, we need to understand how to calculate arc length. So, with our eye on next week’s big prize of discovering the ins-and-outs of the nautical mile, today we’re going to take a dip into the sea of arc lengths.
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What Is an Arc?
Before we talk about nautical miles, we first need to talk about arc length. And before we talk about arc length, we need to talk about arcs. So, what is an arc? There’s a good chance you already know the answer to this question, but just in case you don’t, here’s a quick refresher.
To keep things simple, today we’re mostly going to be talking about what are called “circular arcs.” A circular arc is simply a portion of the circumference of a circle; in other words, if you draw a circle and then erase part of it (as much or as little as you’d like), what you’re left with is an arc.
So who cares about arcs? We all do. After all, humans really love circles. And we end up making lots of things out of pieces of them—that is, out of the arcs. I’m thinking about curves in roads, the curved end of the pizza cutter sitting in front of me right now (it was a delicious meal), and on and on.
We are surrounded by circles and their arc-shaped offspring. As such, we’d better learn how to calculate the length of an arc—also known as the arc length—don’t you think?
What Is Arc Length?
If you want to find the length of a straight line, you can get out a ruler and measure it. But what if you’re trying to find the length of a circuitous road? That’s a little harder.
The problem of finding the length of an irregular arc vexed smart people for thousands of years.
In fact, the problem of finding the length of an irregular arc (as opposed to a circular arc) vexed smart people for thousands of years. It wasn’t until the invention of calculus a couple hundred years ago that mathematicians were really able to solve the problem.
Just for fun, let’s imagine how you might go about measuring the length of a squiggly road on a map. One thing you could do is take a flexible piece of string, lay it out along the road, mark the beginning and end points on the string, and then stretch the string back out and measure the now-straightened length of the road with a ruler.
But what if you don’t have a flexible piece of string—what else could you do?
Whether or not you’ve realized it, you already know how to calculate circular arc lengths – at least, as long as you know the famous formula for the circumference of a circle.
The best approach is to approximate the curvy road as a sequence of small straight-line segments. The length of the squiggly road is roughly equal to the sum of the lengths of these straight-line segments.
And if you divide the road up into more and more of these lines, and make them shorter and shorter, your approximation becomes more and more accurate. Keep making them smaller and smaller until you end up with many infinitely short straight lines, and the sum becomes the true length of the arc—this is the essence of the solution provided by calculus.
Fortunately for us though, we don’t have to worry about any of this, because things are much simpler when it comes to finding the lengths of circular arcs.
How to Calculate Circular Arc Length
In fact, whether or not you’ve realized it, you already know how to calculate circular arc lengths…at least, as long as you know the famous formula for the circumference of a circle:
circumference = 2π • r
How is this formula helpful here? Well, think about it: what is the circumference, really? It’s the distance around the circle. In other words, it’s the length of the arc that is the circle. So the circumference is actually an arc length!
As we learned when discussing radians and degrees, 2π radians is the angle around the entire circle (it’s the same as 360 degrees.) Which means that the formula for the circumference of a circle is really just a special case of the formula for the arc length,
arc length = θ • r
In this formula, r represents the radius of the circle, and θ represents the angle from one end of the circular arc to the other in radians. The only difference between this formula and the one for the circumference is that the angle can be anything in order to represent any arc length, not just 2π.
What Is Arc Length Useful For?
As we talked about at the outset, this whole idea of arc length turns out to be extremely important in the real world. As we’ll see next time, one way in which it’s used is in the definition of the so-called nautical mile. To tide you over, here’s a little arc length inspired puzzle for you to think about that has plenty of real world use potential.
The problem is this: Imagine you have a pizza in front of you, from which you consume a large pie-shaped wedge. This kind of pizza-shaped wedge taken out of a circle is known as a “sector of the circle.”
If the sector you eat spans an angle θ, and if the pizza has a radius of r, what’s the area of the pizza you’ve eaten in terms of that angle and radius? Here’s a hint: It’s got something to do with arc length. (Big surprise!) Think about it, and be sure to check back next time for the answer.
Wrap Up
Okay, that’s all the 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!
Arcs image courtesy of Shutterstock.
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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.