0:00:00.075,0:00:01.981
- [Voiceover] What I want[br]to talk about in this video,

0:00:01.981,0:00:04.788
is the notion of Arc Measure,

0:00:04.788,0:00:06.946
when we're dealing with circles.

0:00:06.946,0:00:09.175
As we'll see, sometimes when you see

0:00:09.175,0:00:10.499
something like arc measure,

0:00:10.499,0:00:12.356
you might think it's the length of an arc,

0:00:12.356,0:00:14.771
but arc length is[br]actually a different idea.

0:00:14.771,0:00:16.884
So we will compare these two things.

0:00:16.884,0:00:20.297
Arc length to arc measure.

0:00:20.297,0:00:23.269
So arc measure, all that is

0:00:23.269,0:00:24.685
is just a fancy way of saying,

0:00:24.685,0:00:27.077
if I have a circle right over here,

0:00:27.077,0:00:29.933
this is my best attempt[br]at drawing a circle.

0:00:29.933,0:00:32.557
I have a circle here.

0:00:32.557,0:00:37.131
The center of the circle,[br]let's call that point O,

0:00:37.131,0:00:39.731
and let me put some[br]other points over here.

0:00:39.731,0:00:43.725
So let's say that this is point A,

0:00:43.725,0:00:46.837
let's say this is point B,

0:00:46.837,0:00:51.178
and let's say this is[br]point C right over here.

0:00:51.178,0:00:53.222
And let's say that I have,

0:00:53.222,0:00:57.053
let's say the central[br]angle, right over here,

0:00:57.053,0:00:59.166
cause it includes the[br]center of the circle,

0:00:59.166,0:01:02.114
so the central angle, angle AOB.

0:01:02.114,0:01:07.078
Let's say it has a measure of 120 degrees.

0:01:09.638,0:01:11.077
And if someone were to say,

0:01:11.077,0:01:14.345
what is the measure of arc AB?

0:01:14.769,0:01:16.324
So, let me write that down.

0:01:16.324,0:01:17.600
The measure.

0:01:17.600,0:01:22.476
So, if someone were to say[br]what is the measure of arc AB,

0:01:22.476,0:01:23.916
and they'd write it like this,

0:01:23.916,0:01:27.444
so that's referring to[br]arc AB right over here.

0:01:27.444,0:01:31.414
It's the minor arc, so there's[br]two ways to connect AB,

0:01:31.414,0:01:33.481
you could connect it right over here,

0:01:33.481,0:01:35.408
this is the shorter distance,

0:01:35.408,0:01:36.942
or you can go the other way around,

0:01:36.942,0:01:39.843
which would be what you'd[br]consider the major arc.

0:01:39.843,0:01:41.840
Now, if someone's[br]referring to the major arc,

0:01:41.840,0:01:44.556
they would say arc ACB.

0:01:44.556,0:01:46.809
So when you're given just two letters,

0:01:46.809,0:01:49.480
you assume it's the shortest[br]distance between the two.

0:01:49.480,0:01:51.661
You assume that it is the minor arc.

0:01:51.661,0:01:53.473
In order to specify the major arc,

0:01:53.473,0:01:55.423
you would give the third letter,

0:01:55.423,0:01:57.373
to go the long way around.

0:01:57.373,0:02:00.646
So the measure of arc AB, and sometimes

0:02:00.646,0:02:02.639
you'll see it with[br]parenthesis right over here,

0:02:02.639,0:02:05.797
all this is, this is the[br]same thing as the measure

0:02:05.797,0:02:10.185
of the central angle[br]that intercepts that arc.

0:02:10.185,0:02:12.670
Well, the central angle[br]that intercepts that arc

0:02:12.670,0:02:16.315
has a measure of 120 degrees.

0:02:16.315,0:02:20.425
So this is just going to be 120 degrees.

0:02:20.425,0:02:22.004
Now, some of y'all might be saying,

0:02:22.004,0:02:23.514
well, what about the major arc?

0:02:23.514,0:02:24.813
Well, let's write that.

0:02:24.813,0:02:27.274
So if we're talking about arc ACB,

0:02:27.274,0:02:29.271
so we're going the other way around,

0:02:29.271,0:02:31.548
so this is major arc.

0:02:31.548,0:02:36.504
So what is the measure of arc ACB,

0:02:36.864,0:02:38.652
once again we're using three letters,

0:02:38.652,0:02:41.067
so that we're specifying the major arc.

0:02:41.067,0:02:45.130
Well, this angle, this[br]central angle right over here,

0:02:45.130,0:02:48.868
to go all the way around[br]the circle is 360 degrees.

0:02:48.868,0:02:52.955
So this is going to be[br]the 360 minus the 120

0:02:52.955,0:02:54.627
that we're not including.

0:02:54.627,0:02:57.273
So 360 degrees minus 120 is going to be,

0:02:57.273,0:03:01.592
is going to be 240 degrees.

0:03:01.592,0:03:06.556
So the measure of this angle[br]right over here is 240 degrees,

0:03:06.974,0:03:11.803
so the measure of this arc,

0:03:11.803,0:03:14.055
I have to be careful not[br]to say length of that arc,

0:03:14.055,0:03:16.192
the measure of this arc[br]is going to be the same

0:03:16.192,0:03:18.026
as the measure of the central angle.

0:03:18.026,0:03:21.694
It's going to be 240 degrees.

0:03:24.424,0:03:27.476
These arc measures are[br]going to be the case

0:03:27.476,0:03:29.659
regardless of the size of the circle,

0:03:29.659,0:03:31.376
and that's where the[br]difference starts to be

0:03:31.376,0:03:33.513
from arc measure to arc length.

0:03:33.513,0:03:36.764
So, I could have two circles,

0:03:36.764,0:03:38.900
so this circle right over here

0:03:38.900,0:03:42.033
and that circle right over here,

0:03:42.033,0:03:45.609
and as long as the central[br]angle that intercepts the arc

0:03:45.609,0:03:47.885
has the same degree measure,

0:03:47.885,0:03:52.885
so let's say that that degree[br]measure is the same as,

0:03:53.025,0:03:54.357
these are central angles,

0:03:54.357,0:03:55.844
so we're assuming the vertex of the angle

0:03:55.844,0:03:57.284
is the center of the circle.

0:03:57.284,0:03:59.559
As long as these two are the same,

0:03:59.559,0:04:02.253
these two central angles[br]have the same degree measure,

0:04:02.253,0:04:05.457
then the arc measures, then[br]the corresponding arc measures

0:04:05.457,0:04:07.012
are going to be the same.

0:04:07.012,0:04:10.402
But clearly, these two[br]arc lengths are different.

0:04:10.402,0:04:12.260
The arc length is not going to depend

0:04:12.260,0:04:15.163
only on the measure of the central angle,

0:04:15.163,0:04:16.974
the arc length is going to depend

0:04:16.974,0:04:20.132
on the size of the actual circle.

0:04:20.132,0:04:22.825
Arc measure is only dependent

0:04:22.825,0:04:26.052
on the measure of the central angle

0:04:26.052,0:04:28.885
that intercepts that arc.

0:04:28.885,0:04:33.087
So your maximum arc measure[br]is going to be 360 degrees.

0:04:33.087,0:04:36.733
Your minimum arc measure is[br]going to be zero degrees.

0:04:36.733,0:04:39.984
It's measured in degrees,[br]not in units of length

0:04:39.984,0:04:42.561
that arc length would be measured in.

0:04:42.561,0:04:44.278
So, let me write this down.

0:04:44.278,0:04:46.275
This only depends...

0:04:46.275,0:04:49.387
So, this is what's going to drive this

0:04:49.387,0:04:54.387
is the measure of central[br]angle, central angle,

0:04:57.798,0:05:00.045
that intercepts the arc.

0:05:00.747,0:05:05.747
That intercepts, intercepts the arc.

0:05:06.757,0:05:08.684
When you talk about arc length,

0:05:08.684,0:05:10.797
yes, it's going to be[br]dependent on the angle,

0:05:10.797,0:05:12.747
but it's also dependent,[br]it's going to be dependent

0:05:12.747,0:05:14.233
on the measure of that central angle

0:05:14.233,0:05:16.875
plus the size of the circle.

0:05:16.875,0:05:19.452
Size of the circle.

0:05:19.452,0:05:21.519
You're actually talking[br]about a length now,

0:05:21.519,0:05:22.935
when you're talking about arc length.

0:05:22.935,0:05:25.298
While here, you're talking[br]about a degree measure.