If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Geometry (all content)>Unit 14

Lesson 2: Arc measure

# Intro to arc measure

Sal introduces arc measure including that they must sum to 360, conventions of minor and major arcs, and a comparison to arc length.

## Want to join the conversation?

• at how do you know its 240 degrees? and how can you solve equations using these methods
• Think about what you just posted and revise it before I call your teacher
• If the arc measure is 180° then which would be the minor/major arc or there would be no minor or major arc
• Great question! I was actually curious about that myself and what I've found is that a major arc is always greater than 180° and the minor arc is always less than 180°.

This means that there would be no minor/major arc, but instead both halves of the circle would be called semicircles.

Hope this helps,
- Convenient Colleague
• Two points lying on a circle actually define two arcs. The shortest is called the 'minor arc' the longer one is called the 'major arc'.A Major arc is usually referred with three letters, and a minor arc is usually referred to with only two letters. Typically, if you don't specify which arc you mean, then most people will assume you mean the minor (shortest) arc. If there is a possibility of confusion, you should state which one you mean. in this video, Sal probably referred to the minor arc with three letters because he had a picture to clarify. if an angle is 180 degrees, then it is a major arc.
• I think Doug was asking what if there was no third point listed? I had the same thought.
• When an Arc is described as 2 letters, which type of an Arc?
When an Arc is described as 3 letters, which type of an Arc?
• 2 letters is a minor arc (less than 180 degrees) and 3 letters is a major arc (greater than 180 degrees)
• I am wondering, when I take the test, I do not know whether they want me to find the minor arc or, the minor arc subtracted from the circle(360 degrees)
Do you know how to tell what they want? Different in wording or something?
• As Sal says in the video at around , you would usually use two letters for a minor arc and three letters to describe a major arc. Hope this helps!
• Is a 180 degree central angle a major or minor angle? And if the major angle has only 2 letters how should you name it?
• It depends on how the angle is named so 180 degree would be major if it was named with 3 points using the example in the video measure ACB but it would be minor if it only used two letters example AB. If the angle only has two letters then it is not major it is minor. REMEMBER it is the number of letters - 3 for major 2 for minor.
• Why is the arc measure the same as the central angle if it it so much bigger than it
• The arc measure is just the measure of how much the arc is around the circle. It is literally the same thing as the central angle because they both describe the same thing.

The arc length is the length of the arc and it will the distance of the arc which is what you were probably thinking about.

Hope this helps,
- Convenient Colleague
• how would u write it if says arc AOB?
• There is no such thing as arc AOB because he is using O as the center of the circle. Throughout the video, he says AOB is an angle, so you could write an angle sign and AOB after it. To signify atc AB, you write the letters and draw an arc symbol above it. He does this at about .
• At , what does intercept mean in the context of Mr. Khan's sentence?
• An angle intercepts an arc when it "captures" it (just like when a football player makes an interception). An angle captures, or intercepts, an arc when the arc is between the sides of the angle. A central angle intercepts one arc, as does any angle in the interior of the circle. But an angle on the exterior of the circle can intercept two separate arcs.
• If an arc has a measure of 180 degrees, is it a minor or a major arc or does it just have some other name?
• It is a semicircle. Semi is a prefix meaning 1/2.

## Video transcript

- [Voiceover] What I want to talk about in this video, is the notion of Arc Measure, when we're dealing with circles. As we'll see, sometimes when you see something like arc measure, you might think it's the length of an arc, but arc length is actually a different idea. So we will compare these two things. Arc length to arc measure. So arc measure, all that is is just a fancy way of saying, if I have a circle right over here, this is my best attempt at drawing a circle. I have a circle here. The center of the circle, let's call that point O, and let me put some other points over here. So let's say that this is point A, let's say this is point B, and let's say this is point C right over here. And let's say that I have, let's say the central angle, right over here, cause it includes the center of the circle, so the central angle, angle AOB. Let's say it has a measure of 120 degrees. And if someone were to say, what is the measure of arc AB? So, let me write that down. The measure. So, if someone were to say what is the measure of arc AB, and they'd write it like this, so that's referring to arc AB right over here. It's the minor arc, so there's two ways to connect AB, you could connect it right over here, this is the shorter distance, or you can go the other way around, which would be what you'd consider the major arc. Now, if someone's referring to the major arc, they would say arc ACB. So when you're given just two letters, you assume it's the shortest distance between the two. You assume that it is the minor arc. In order to specify the major arc, you would give the third letter, to go the long way around. So the measure of arc AB, and sometimes you'll see it with parenthesis right over here, all this is, this is the same thing as the measure of the central angle that intercepts that arc. Well, the central angle that intercepts that arc has a measure of 120 degrees. So this is just going to be 120 degrees. Now, some of y'all might be saying, well, what about the major arc? Well, let's write that. So if we're talking about arc ACB, so we're going the other way around, so this is major arc. So what is the measure of arc ACB, once again we're using three letters, so that we're specifying the major arc. Well, this angle, this central angle right over here, to go all the way around the circle is 360 degrees. So this is going to be the 360 minus the 120 that we're not including. So 360 degrees minus 120 is going to be, is going to be 240 degrees. So the measure of this angle right over here is 240 degrees, so the measure of this arc, I have to be careful not to say length of that arc, the measure of this arc is going to be the same as the measure of the central angle. It's going to be 240 degrees. These arc measures are going to be the case regardless of the size of the circle, and that's where the difference starts to be from arc measure to arc length. So, I could have two circles, so this circle right over here and that circle right over here, and as long as the central angle that intercepts the arc has the same degree measure, so let's say that that degree measure is the same as, these are central angles, so we're assuming the vertex of the angle is the center of the circle. As long as these two are the same, these two central angles have the same degree measure, then the arc measures, then the corresponding arc measures are going to be the same. But clearly, these two arc lengths are different. The arc length is not going to depend only on the measure of the central angle, the arc length is going to depend on the size of the actual circle. Arc measure is only dependent on the measure of the central angle that intercepts that arc. So your maximum arc measure is going to be 360 degrees. Your minimum arc measure is going to be zero degrees. It's measured in degrees, not in units of length that arc length would be measured in. So, let me write this down. This only depends... So, this is what's going to drive this is the measure of central angle, central angle, that intercepts the arc. That intercepts, intercepts the arc. When you talk about arc length, yes, it's going to be dependent on the angle, but it's also dependent, it's going to be dependent on the measure of that central angle plus the size of the circle. Size of the circle. You're actually talking about a length now, when you're talking about arc length. While here, you're talking about a degree measure.