Arc Length and Circumference
This figure might represent you and a friend running around a circular track—you in the inside lane and your friend in the outside lane. If you both complete the same fraction of the track (say, a quarter-lap), does that mean you both run the same distance?
We have two arcs, and , intercepted by the same central angle. By definition, arcs and have the same measure. But they don't look like they have equal length, do they? We can solve this mystery, but we need more clues. Even Sherlock Holmes couldn't work with nothing.
The arc length of a circle (that is, the distance a bug would have to crawl to go around the circle exactly once, staying on the circle the whole time) is called its circumference.
The ratio of a circle's circumference to twice its radius is a constant, which we represent with the Greek letter π ("pi," pronounced "pie," like the circle-based dessert).
In symbols, this gives us the formula
C = 2πr
C is the circumference of the circle and r is its radius. The constant π is an irrational number (but nobody's perfect). We can't represent it exactly as a ratio of two integers, and if we write it in decimal form we find that it's an infinite, non-repeating decimal. We can use approximations: π is approximately equal to 3.14, or 22/7. Those approximations get the job done pretty well. However, if we want to be exact, we have to simply write π.
What is the circumference of a circle with radius r = 7 inches?
All we need in order to calculate the circumference of a circle is the radius. Well, that and the formula. Luckily, we have both. We can substitute 7 for r in the formula C = 2πr to end up with C = 2π(7) = 14π ≈ 44 inches.
Now we know how to find the circumference of a circle given its radius. We also know how to find the length of an arc with measure 360° given its radius, because that's the same thing as a circle. Two birds with one stone.
How about a semicircle? Can we find the length of a semicircle, given its radius? Spoiler alert: yes. It ought to be half the circumference of a circle with the same radius, don't you think? And let's say we wanted to find the length of a quarter-circle with a given radius. Shouldn't it be one-fourth of the circumference of a circle with the same radius?
We can generalize that idea like this: The length L of any arc with radius r and measure θ can be expressed as follows:
The above formula asks us, "How much of a circle is the arc?" and "How big is the circle?" and gives us the length of the arc. This will be a powerful tool in figuring out the Case of the Unfair Track Run.
Suppose a circular track has a radius of 50 m and the distance between the inner lane and outer lane is 10 m. If you ran a quarter-lap along the inner lane and your friend ran a quarter-lap along the outer lane, what's the length that you ran? How far did your friend run?
Since this is an arc length problem, we know we'll need to use the arc length formula. That means we need to know θ and r for both tracks. Since you and your friend both run one-fourth of a lap, that translates to θ = 360° ÷ 4 = 90°.
The radii of the tracks, on the other hand, are different. You run on the inner track, which has a radius of 50 m. Your friend runs on the outer track, which has a radius of 50 + 10 = 60 m. Plugging those values into the formula, we know that you ran about 78.5 m, while your friend ran 94.2 m.
Your friend ran a greater distance than you, even though you both ran through the same angle. In other words, you and your friend ran arcs of different lengths, even though they had the same measure. A good mathematician has the power to be a bad friend. But you can always offer your friend some pi to make up for it.