*Jeff Calareso*Show bio

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Lesson Transcript

Instructor:
*Jeff Calareso*
Show bio

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

The measure of an arc can be solved by determining the degree of the angle that intercepts an arc. Learn about the process of measuring an arc, major vs. minor arcs, and real-world practice arc problems.
Updated: 09/28/2021

Depending on where you live, you might encounter different kinds of arcs. If you have a neighbor named Noah, you might be familiar with this kind of ark. That's 'ark' with a 'k.' By the way, if Noah is your neighbor, you might want to get some flood insurance. If you live in an Indiana Jones movie, you might be familiar with this kind of ark. This is also 'ark' with a 'k.' Also, it has a tendency to kill everyone near it, especially Nazis, so you might want to avert your eyes.

And then there's 'arc' with a 'c.' This is the arc of circles. 'Circle' begins with 'c,' so that makes sense. An **arc**, in terms of a circle, is a part of a circle's circumference, or edge. If you prefer your circles in pie form, then an arc is the flaky crust keeping in that delicious pumpkin filling on your slice. I know I like my vegetables in pie form, and I love me some pumpkin pie.

If you're like me, then when you cut a slice of pie, you have to stop and think about the two arcs you just created. Your knife makes two points on the circle. Let's call them A and B. If we focus on the normal slice, we call that the **minor arc**, or the shorter arc joining two points on a circle's edge. That's arc AB.

And then there's the **major arc**, or the longer arc joining two points on a circle's edge. You could think of this in terms of major and minor leagues in baseball. Major leagues are the big time, so major arc is the bigger one.

But wait. Do we also call that arc AB? We can. But that could mean the major or minor arc. So sometimes we just say major arc AB. Or we add a point like this. Then the major arc is arc AQB. As for the pie, you could totally take that gigantic major arc slice. But you resist, right?

Now maybe you're comparing slices of this key lime pie and this key lime tart. No matter what, this counts as a fruit, right? Anyway, this is where we differentiate between an arc's measure and its length. The **measure of an arc** is the degree measurement of the angle that intercepts the arc. The measure of an arc is written as mAB.

Notice that I said it's the degree measurement. That's important. And it's different than an arc's length, which is the distance along the circumference of the arc. Look at our comparable pie and tart slices. The length of the arc on the pie slice is a few inches. On the tart? Barely an inch. But the measure of these arcs is the same. It's based on the degree of this angle, the tip of our piece.

What's the total of the degrees inside a circle? 360. So in half the pie, it's 180 degrees. In a quarter of the pie, you can see that that's a right angle, and it's 90 degrees. That makes sense, since it's a quarter of the pie, and a quarter of 360 is 90. In our pie and tart, both arcs measure 42 degrees. So the lengths can vary based on the circle's size, but the measure remains the same.

Let's look at a few practice pies. Here's a Dutch apple - that's one of my favorites. If the minor arc, the normal person slice, measures 40 degrees, what is the measure of the major arc? It's just 360 - 40, or 320. Here's a cherry pie, another one of my favorites. We get to this pie a little late. Someone already cut out a 50 degree arc and a 45 degree arc. What's the measure of the remaining arc? It's just 360 - 50 - 45, which is 265. That leaves us plenty of pie.

Here's a pecan pie, also a favorite of mine. Okay, I really like pie. If we're trying to share this pie, and we cut an arc with a measure of 175 degrees, did we leave a major or minor arc? Remember, the major arc is the larger arc. So since the remaining arc is 185 degrees, which is larger than 175, it's the major arc.

In summary, an arc is a part of a circle's edge, or circumference. When two points are identified on a circle, the shorter arc is the minor arc and the larger one is the major arc. The measure of an arc is the degree of the angle that intercepts the arc. Remember that the measure of an arc is a degree, not a distance. We write the measure of an arc as mAB. To differentiate between major and minor arcs, we either say major arc AB or we add another point, so we know that mAQB is the major arc. Now I think it's time for some pie.

After this lesson, you should be able to:

- Identify an arc
- Determine minor arcs from major arcs
- Understand how to measure an arc

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