What Is the Circumference of a Circle?

Throughout history, circles have symbolized many things: unity, protection, the Sun, infinity, and the Olympic Games, to name a few. Of course, philosophers and symbologists aren’t the only people to have taken an interest in circles. Mathematicians have spent millennia studying them, too. Which is precisely why today’s article is all about circles. In particular, after a quick refresher of circle basics, we’re going to figure out why the equation for the circumference of a circle that we all learned in school works, and we’re also going to learn how this equation is used to set up lots of Olympic track and field events.

What Is a Circle?

My favorite way to define a circle is in terms of how to draw one. In the episode What Is Pi? we learned how to draw a circle arts-and-crafts style. To do this, start by cutting a 3-inch piece of string to serve as the radius of the circle. What’s the radius? It’s half the diameter. Okay, but what’s the diameter? As you’ll soon be able to test with your finished drawing, it’s the greatest distance between any two points on the circle. Now, pin one end of the string down with your finger near the center of a normal sheet of binder paper, and then hold the loose end of the string up against the lead of your pencil. Finally, pull the pencil so the string is taut and trace out your circle. 

What does this all mean? Believe it or not, it means we’ve found a very good way to define a circle. Namely, a circle is the set of all points (that’s the curve you drew with your pencil) that are all the same distance from some common point (that’s the spot where you pinned the string down with your finger).

How to Calculate the Circumference of a Circle

Now that we have a circle to work with, let’s investigate its circumference—aka, the distance around the outside of the circle. We could start this investigation by laying a long piece of string all the way around our 3-inch radius circle, cutting it so that the string is the same length as the circle’s circumference, and then laying this string out in a line and measuring its length with a ruler. 

I certainly encourage you to carry out that little investigation if you want to, but the beauty of math is that we really don’t need to. Why? Because of the way that the number pi is defined.
As we’ve talked about before, the number π = 3.14… is defined as the ratio of the circumference of a circle to its diameter. In other words, π = C / d.
This means that to find the circumference of any circle, we just have to rearrange this equation: C = π × d.
Since we know that the diameter of a circle is just twice its radius, we can also express the formula as: C = 2 × π × r.
So, to find the circumference of our 3-inch radius circle, we apply this formula: C = 2 × 3.14 × 3 inches, which gives us approximately 18.84 inches.

How Long Are the Olympic Rings?

So we now know how to calculate the circumference of a circle. But so what? Why is that important? Well, there are lots of reasons that you might want to make this calculation. For example, what if you wanted to use colored yarn to put together a giant tapestry of the 5 interlocking Olympic rings for your bedroom wall? If you were so inclined, all you’d need to know to find out how much yarn you’d need is the radius of your desired rings. If you want 1 foot radius rings, you need C = 2 x 3.14… x 1 foot or about 6.28 feet of string for each ring. Obviously this isn’t something most of us are ever going to want to do, which might leave you wondering if there are any real world applications of our famous formula?

Circles in Olympic Track and Field Events

If you’re looking for a great real world example of the use of the formula for the circumference of a circle, look no further than Olympic track and field. In particular, have you ever noticed how the runners in some track and field races start at different places on the track? If you think about it for a minute, it’s pretty easy to figure out that this type of staggered starting is necessary to compensate for the differences in how far the runners in different lanes have to run when going around curves. 

So, how do the people laying out the track figure out where to draw the different starting lines? Well, if you look at a track, you’ll see that it’s really just made of 2 semi-circles connected to 2 straightaways. If you measure the radius of the semi-circle made by each lane of the track, you can use the circumference formula to calculate the distances that the runners in each lane travel. The farther out you are, the longer the distance you have to run to make it around the track—which is precisely why the runner in the outside lane starts in front, followed by the runner in the next most outside lane, then the next, and so on. And you can calculate exactly what the distances between each runner should be at the start just by applying the formula for the circumference of a circle.

Bonus Tip: How to Easily Convert From KPH to MPH

Before we finish up for today, I want to share a quick and dirty tip that math fan Dan sent me in response my tip in last week’s Olympic-themed episode about converting from kilometers to miles (and kilometers per hour to miles per hour) in your head. Dan points out that it’s kind of tough to divide by 1.6 in your head as I suggested. And he’s definitely right—it’s fairly easy to multiply numbers in your head, but doing division is tough. 

So Dan suggests that instead of dividing by 1.6 to convert from kilometers to miles, we instead multiply by 5/8 (which, as you can check, really is doing the same thing). How is that better? Because multiplying some number by 5/8 is the same as adding 1/2 of that number to 1/4 of 1/2 of that number. Think about it for a few minutes, try it out, and I think you’ll agree with me that Dan’s tip is a good one.

Wrap Up

Okay, that’s all the math we have time for today. Please become a fan of the Math Dude on Facebook where you’ll find lots of great math content posted throughout the week. And if you’re on Twitter, please follow me there too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.

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!