Main content

## High school geometry

# Finding arc measures

CCSS.Math:

Watch Sal solve a few problems where he finds a missing arc measure.

## Want to join the conversation?

- At0:25, isn't the major arc also Arc AC, so wouldn't that be really confusing that both the minor arc and the major arc are called the same thing? Shouldn't the minor arc be called Arc ABC in this case?(21 votes)
- Hi Hisham Malik,

The major arc has to be denoted with three letters so it will be called arc ACB. The minor arc is always two letters so in this case it is AC. Always remember that a major arc is always 180˙ or more.

Hope that helps!

- JK(16 votes)

- Wait, so Sal means that the angle value is the same as the arc measure? I thought they were two different things. My math teacher said so. Please help.(13 votes)
- The arc measure is equal to the angle value. It's just like taking a protractor to those two lines. However, the arc LENGTH is different. The arc length would be like cutting that piece of the circle off and measuring it with a ruler, therefore it is measured in inches, mm, etc.(7 votes)

- What if an arc is exactly 180 degrees? Is it a major or minor arc? (Sorry if this is a stupid question :P)(10 votes)
- An arc that is exactly 180 degrees is a semicircle. Also, that's actually a really good question.(13 votes)

- i think the first example was poorly phrased, wouldn't the correct answer be 186 degrees because you're looking for arc AC instead of ABC? typically, the major arc contains three points on the circle but in this scenario the minor arc does, so how do you determine which arc you're looking for? asking for a friend(6 votes)
- Not at all. The minor arc only needs the two endpoints to identify it, there could be as many points in between these as you want (in this case only one), it does not change the name of it. Major arcs must have three letters to distinguish them from minor arcs, so there would have to be another point D on the opposite side of the circle from B to distinguish it as major arc ADC.(7 votes)

- For the second example, the question says that both AD and CE are diameters of circle P, and I was a bit confused because if both of them are diameters, wouldn't that split the circle into fourths that all equal 90 degrees? How would angle EPD equal 93 degrees when the circle is cut by two diameters? Am I missing something?(5 votes)
- Two diameters need not be perpendicular. Being a diameter just means it passes through the center of the circle. It can be rotated any angle.(5 votes)

- Is being a minor arc a bad thing or a good thing?(2 votes)
- It actually basically doesn't basically technically essentially matter at all. So there you go! :-)(4 votes)

- For the first question if arc AC is the minor arc, then what would be the major arc?(3 votes)
- The major arc (or the longer arc, which must be larger than 180 degrees) to the first question would be ABC. I hope that this helps you.(0 votes)

- I thought that it would be major since it takes three angles. So how is it the minor?(1 vote)
- How can you tell which arc it is talking about? For example, Arc AE is also Arc EA or when there are only two labeled points(1 vote)
- You are correct that arc AE and arc EA are the same, but arc ACE is different. If you have an arc with just two points, you know it is a minor arc (less than 180 degrees) and if you have three points, you know it is a major arc (more than 180 degrees), so the number of points tells you which way to go between the two end points.(4 votes)

- So in the first problem, where <ApB = 70 and <BpC = 104 how would you specify the MAJOR arc? This concept of symbology seems very poorly conceived. Sal say we must ASSUME two letters refer to the minor arc, but there is no third letter available to specify the MAJOR arc. Pretty poor assumption, in my humble opinion.(2 votes)
- It is really simple.

If they want you to find the major arc, then they can specify it in a couple of ways:

1. They can say Major arc AC

2. Or, they can say the minor arc is arc ABC and the major arc is just AC. If the B is absent in one, then we can assume that B is not part of the arc.(1 vote)

## Video transcript

- So I have some example questions here from Khan Academy on arc measure. And like always, I encourage
you to pause the video after you see each of these questions, and try to solve them before I do. So this first question says
what is the arc measure, in degrees, of arc AC on circle P below. So this is point A, that is point C, and when they're talking about arc AC, since they only have two letters here, we can assume that it's
going to be the minor arc. When we talk about the minor arc. There's two potential arcs that
connect point A and point C. There's the one here on the left, and then there's the one, there is the one on the right. And since C isn't exactly
straight down from A, it's a little bit to the right, the shorter arc, the arc with the smaller length, or the minor arc is going to be this one that I'm depicting here
right on the right. So what is this arc measure going to be? Well, the measure of
this arc is going to be exactly the same thing as, in degrees, as the measure of the central
angle that intercepts the arc. So that central angle, let me
do it in a different color, I'll do it in this blue color, that central angle is angle C, P, A. Angle C, P, A, and the
measure of that central angle is going to be 70
degrees plus 104 degrees. It's going to be this whole
thing right over there. So it's going to be 174 degrees. One hundred and seventy four degrees, that's the arc measure,
in degrees, of arc AC. Let's keep doing these. So let me do another one. So, this next one asks us, in the figure below, in the figure below, segment AD-- so this is point A, this is point D, so segment AD is this one right over here. Let me see if I can draw that. That's AD right over there, AD and CE are diameters of the circle. So let me draw CE, so CE is, we're going to connect point C and E. These are diameters. So, let me, so they go straight. Whoops, I'm using the
wrong tool, let me... So those are, somehow I should, alright. So, those are di-- whoops, how did that happen? So let me, somehow my pen got really big, alright. That'll be almost there, ok. So CE, there you go. So those are both
diameters of the circle P. What is the arc measure of
AB, of arc AB in degrees? So arc AB, once again
there's two potential arcs that connect point A and B. There's the minor arc, and since this only has two letters we'll assume it's the minor arc. It's going to be this one over here. There's a major arc, but to not the major arc
they would've said something like A, E, B or A, D, B or arc A, C, B to make us go this kind of, this long way around. But this is arc AB, so we, in order to find the arc measure, we just really have to find the measure of this central angle. This is the central angle
that intercepts that arc, or you can even say it
defines that arc in some way. So how can we figure out this angle? And this one's a little bit trickier. Well, the key to, the key here is to realize
that this 93 degree angle, it is vertical to this
whole angle right over here. And we know from geometry, which we're still learning as
we do this example problem, that vertical angles are going
to have the same measure. So if this one on, this one is 93 degrees, then this entire blue one right over here is also gonna be, let me write it, this is also gonna be 93 degrees. So 93 degrees, that's gonna
be made up of this red angle, that we care about, and the 38 degrees. So this red one, which is the
measure of the central angle, it's also the arc measure of arc AB, is going to be 93 minus, 93 degrees minus 38 degrees. So what is that going to be? Let's see, 93, I can write degrees there, minus 38 degrees, that is going to be equal to, let's see if it was 93 minus 40 it would be, it would be 53, it's gonna be two more, it's gonna be 55 degrees. Fifty five degrees, and we are done. This angle right here is 55 degrees. If you were to add this angle meausre, plus 38 degrees, you would get 93 degrees, and that has the same
measure because it's vertical with this angle right over here, with angle D, P, E. Alright, let's do one more of these. So, we have in the figure below, and it doesn't quite fit on the page, but we'll scroll down in a second, AB is the diameter of circle P, is the diameter of circle P. Alright, so AB is the diameter, let me label that. So AB is the diameter. It's going straight across, straight across the circle. What is the arc measure
of A, B, C in degrees? So A, B, C. So they're making us
go the long way around. This is a major arc they're talking about. Let me draw it. Arc A, what is the arc measure of arc A, B, C. So we're going the long way around. So it's a major arc. So what is that going to be. Well it's going to be in degrees, the same measure as the angle, as the central angle that intercepts it. So it's going to be the same thing as this central angle right over here. Well, what is that
central angle going to be? Well, since we know that
this is the diameter, since AB is the diameter, we know that this part of it is going to 180 degrees. We're going halfway around the circle. One hundred eighty degrees. And so if we wanna look
at this whole angle, the angle that intercepts
the major arc A, B, C, is going to be 180
degrees plus 69 degrees. So we're going to have 180 degrees, plus 69 degrees which is equal to, what is that, 249, 249 degrees. That's the arc measure of
this major arc A, B, C.