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Arc Length - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Commonly confused with arc measure, arc length is the distance between the endpoints along the circle. Arc measure is a degree measurement, equal to the central angle that forms the intercepted arc. Arc length is a fraction of the circumference of the circle and calculated that way: find the circumference of the circle and multiply by the measure of the arc divided by 360.

There are 2 properties when we're talking about circles that are easily confused. Arc length versus arc measure. To compare arc length and arc measure, let's look at some concentric circles. Let's say I drew in 2 radii in the smaller circle and I call that intercepted arc, arc a, let's say I extended those radii all the way out to the larger circle and I'm going to call that, arc b, so again we're trying to compare arc measure with arc length and I see that since it has the same central angle since arc a and arc b have the same central angle, I could say that the measure, the arc measure of a is equal to the measure of b. But if we look at the distance between the intercepted arcs endpoints we can see pretty clearly that the length l, e, n, g, t, h of arc a is definitely smaller than the length of b, so it is possible for 2 arcs to have the same measure but different lengths.
Arc length is the fraction of a circle's circumference so if we looked at a circle over here where I have drawn a right angle and I said what is the distance between x and y? If we know the whole circle has a circumference of c, where c is some circumference, we know the fraction here it's pretty clear is just going to be a fourth of that because 90 degrees is one fourth of the circle so the circle is 360, 90 is one fourth so the way we calculate arc length is if I have some circle with an arc ab, the length of that is the circumference of the whole circle times the fraction of the circle, so this part right here tells you, if you know measure of arc ab just write it over 360 and that will tell you how much of the circle are you using? So for c you could either substitute 2 pi r because that's circumference in terms of radius is equal to the measure of arc ab out of 360 or if you wanted to you can also use pi times diameter times measure of arc ab out of 360. So the key thing here is that arc length is a fraction of the circle circumference, so we're talking about a distance which is why you can have the same arc measure but different arc lengths.

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