Before we get into our super fantastic lesson on the magic of arc length, lets have a small talk about the difference between an arch and an arc. An arc is a segment of a circle, in which all point on said arc are the same distance from the center. An arch is a cool looking thing used in architecture.

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This is a cool arch, but its not a true arc because all its points aren't equidistant from its center, so think about that for a minute.

Wow! Let’s cut a twig off of a tree growing in the wonderful world of super circles today. First off, let me introduce myself! My name is Brian. What’s your name? (5 second pause) That’s a swell name, but I’m just going to call you pal.
Boy oh boy! Circles are pretty cool once you get down the basics. Today, I’ll be learning you on some arc lengths. A guy once asked me, “Brian, what is the deal with arc lengths?” I looked him straight in the eye and I said, “You’re going to need to be way more specific. If you ask me a question that asinine again, just get out.” Fun math anecdotes!
I know what you’re thinking: “Math is for nerds and dorks. I don’t want to learn anything to do with circles.” Well I will tell you what, mister… what is more magical than an infinite locust of points equidistant from a center point?
One may ask one’s self, “Self, what in all of God’s reen goodness is an arc, let alone an arc lengh?” An arc is any unbroken piece of the circumference of a circle. (Whoa! Crazy terminology! I’m lost Brian! Put me back on course.) See, math is fun. Well, I’m going to make this as simple as possible for you pal! You know how a circle is a line that is curved? Well imagine the “lines” are a piece of string and you want to cut off a piece with scissors. The piece you cut of is an ac! The arc is measured by the angle which is formed by the two lines which intersect at the center of the circle from the ends of the arc.
(Hey teach, I have the mathematics of straight lines down pretty good, but how can I find the length of a curved line?) Good question, pal! Maybe if you could sit down, shut up, and pay attention I will tell you. If you know a little bit of info you can solve for arc lengths of a circle. This does not apply to arcs not on a circle. If you know how to find the area under a curve or the lengths of curves and you are not named Mrs. Scherer get out of the classroom right now. Ok, now that we have cleared the room of everyone who is smarter than me except Mrs. Scherer we can begin.
First of all, let’s get a little sign convention system going here. We’ll call the length of the arc “S”. We’ll call the angle of the arc θ (theta). We will also call the length of the radius ‘r’. (But boy that connotation is rally hard!) Are you serious? That’s only 3 letters to remember. Just give me a second to calm down.

Math math math m-m-math math.

Ok, so let’s start off with the degree formula. θ/360= S/2πr.
Whoa, check it out! The ratio of the angle by the total amount of degrees in a circle is equivalent to the arc length divided by the circumference!
In radians it’s even easier! S = rθ
Be careful, if you want to calculate arc lengths with these formulas, the arcs have to come from circles! Also, make sure you already have or can calculate the values of θ and r so you can find S! Here’s some practice problems! Do them yourself and check your answers afterwards. The units of S are the same as the units for r unless directed otherwise.

Degrees:
1.) r:20 cm, θ:90º, s _
2.) r:23 in, θ:23.5º, s _
3.) r:2.87 nautical leagues, θ:42.7º, s _
4.) r:45.39 shakus, θ:233º, s _
5.) r:147,236.4 spats, θ:12.2º, s _
Radians:
1.) r:333 gigaparsecs, θ:3 radians, s _
2.) r:123 mark twains, θ:2.17 radians, s _
3.) r:234 pes, θ:1 radian, s _
4.) r:678 picas, θ:2π radians, s _
5.) r:1 pygme, θ:π, s _
Solutions to problems following this message:

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Math!
Solutions:
If your units are wrong, you are wrong. If your units are wrong you fail.

Degrees:
1.) 31.416 cm
2.) 9.4335 in
3.) 9.5914 nautical leagues
4.) 591.25 shakus
5.) 31,351.071 spats

Radians:
1.) 999 gigaparsecs
2.) 266.91 mark twains
3.) 234 pes
4.) 4260 picas
5.) 3.1416 pygmes

(Yes, those are all real units.)
Still think you're worthless at arc lengths? Want a program to do the work for you because you're too lazy to do math? This is a recipe I cooked up late Thursday night just before the project was due the next day with my limited computer skills. It does all the work for you! Just type in the r and θ value. (It gives the answer if you put in θ in degrees or radians)