Video: Central Angles in a Circle

In this video, we will learn how to identify a central angle in a circle and find its measure using its properties.

15:07

Video Transcript

In this video, we will learn how to identify a central angle in a circle and find its measure using its properties. To illustrate this, let’s think about Jack. The track he’s running on is 400 meters all the way around. And let’s say Jack runs one-fourth of the track. If we highlight one-fourth, it would be this space. And then if we imagine Jack’s coach is standing in the middle of the track, if he follows Jack around without moving from the center, we see that he has done one-fourth a turn. And a one-fourth turn we know to be a 90-degree turn. If we want to list the distance Jack ran, we could say that he ran 100 meters because 100 is one-fourth of 400.

Now, let’s add to this situation that there is a larger track outside the smaller track. The larger track is 600 meters all the way around. And Jill runs one-fourth of this track. If we highlight one-fourth of this track, it would be this space. We should notice something interesting. The coach in the middle still makes one-fourth of a turn as he watches Jill. The coach does the same rotation for both Jack and Jill. But Jill did not run 100 meters. To find out how far she ran, we would need to take one-fourth of 600 meters, which is 150.

From this, we can draw some conclusions. We see that the distance Jack ran over the total distance of the track is in the relationship of one to four. And the same thing is true for Jill. The distance she ran over the total distance of the track is in the proportion of one over four.

And now we wanna see if we can write a proportion for this turn that the coach made while watching them run. We said he made a one-fourth turn, a 90-degree turn. The 90 degrees represents the amount that Jack and Jill ran. And if that’s the case, what would the whole distance of the track represent?

Starting at the beginning and making a full turn we know would be 360 degrees. And so a full turn is 360 degrees. The coach turned 90 degrees out of 360 degrees, which is in the same proportion as the distance Jack and Jill ran. What we’re seeing here is there is a proportional relationship between the angle at the center and its opposite arc.

So let’s flesh that out a little bit more. If we have a circle with a center 𝑀 and points 𝐴 and 𝐵 fall on the circle, then both 𝐴𝑀 and 𝑀𝐵 are a radius of the circle. The curve between the points 𝐴 and 𝐵 is called arc 𝐴𝐵. We sometimes write the two letters with a curve over the top, which we read arc 𝐴𝐵. The angle created by the points 𝐴, 𝑀, and 𝐵 is angle 𝐴𝑀𝐵, and it is a central angle of this circle. The angle subtended at the center of the circle by two given points on the circle is a central angle. This means if we have another two points on the outside of the circle, 𝐶 and 𝐷, which creates arc 𝐶𝐷, then angle 𝐷𝑀𝐶 would be another central angle of this circle.

Before we move on, we need to make a clarification. For this circle, we would say that angle 𝐷𝑀𝐶 is a central angle. And we have a little indicator about which angle this is. If you didn’t have that and you just had that angle 𝐷𝑀𝐶 is a central angle, you might wonder which of these two angles would be the central angle. And to that we can add that the central angle is the one that is less than or equal to 180 degrees. The smaller of the two angles is the central angle. The larger angle 𝐷𝑀𝐶 is called a reflex angle. And that means when we’re identifying central angles, we want to choose the one that is equal to or less than 180 degrees.

Before we look at some examples, we want to see how we find the measure of a central angle, given other information about the circle. To find the measure of a central angle, we need to think of this proportion. The central angle over 360 degrees is equal to the arc length of that central angle over the circumference of the circle. This means if you know the arc length and the circumference of the circle, you can find its central angle. It also means if you know the central angle and the circumference of the circle, you could find the arc length. Or if you knew the central angle and the arc length, you could find the circumference.

Of course, this is when we are working in degrees. If we want to work in radians, we need to use the central angle over two 𝜋 is equal to the arc length over the circumference. This is because a full turn in radians is two 𝜋.

If we think about Jack again and his 400-meter track, this time he ran 120 meters. The distance that the coach turned watching him is the central angle. We might label this angle 𝜃. And if we wanna know in degrees what this angle is, we can say the central angle 𝜃 over 360 degrees equals the arc length of 120 meters over the circumference of 400. That’s the distance all the way around.

We can solve this a few different ways. If we divide 120 by 400, it’s three-tenths. 𝜃 over 360 degrees needs to equal three-tenths. So we multiply both sides of the equation by 360, and we get 𝜃 equals 108 degrees. The central angle here, the turn that the coach made, was 108 degrees.

Now we’re ready to consider some examples.

Jacob turns one-third of a full turn. How many degrees is this?

To answer this question, let’s think about the phrase one-third of a full turn. If we write one-third, and we know mathematically the word “of” means multiply, if Jacob turns one-third of a full turn, then the key here is knowing how much a full turn is. We’re solving in degrees, which means we want to know how many degrees is a full turn. If we think of one full rotation, it will be 360 degrees. If one full turn is 360 degrees, one-third of that will be the turn that Jacob made. One-third times 360 is 120 degrees. If we want to include this in the image, this angle measures 120 degrees.

In our next example, we’ll be given a circumference and an arc length and we’ll need to use that to find a central angle.

In the figure, given that the circumference is 96 and that the arc length of 𝐴𝐵 is 12, find 𝜃.

When we look at the figure, we see that 𝜃 is a central angle that is subtended by arc 𝐴𝐵. And we remember that a central angle over 360 degrees is in the same proportion as the arc length over the circumference. We know that 𝜃 is our central angle and that a full turn in a circle is 360 degrees. If the arc length is 12 and the circumference is 96, we now have a proportion we can solve for. 12 divided by 96 is 0.125.

To solve for 𝜃, we need to get 𝜃 by itself. We can do that by multiplying both sides of the equation by 360 degrees, which leaves us with 𝜃 on the left. And 0.125 times 360 is equal to 45 degrees.

Another way we could’ve solved this is by first simplifying 12 over 96. I know that 12 goes into 96 eight times. And that means 𝜃 over 360 degrees is equal to one over eight. From there, we would still need to multiply both sides of the equation by 360 degrees. And we would say 360 over eight. 360 divided by eight equals 45 degrees. Both methods show the central angle subtended by arc 𝐴𝐵 to be 45 degrees.

In our next example, we’ll think about a 45-degree angle in relation to a full turn.

How many 45-degree angles does it take to make a full turn?

First, we need to think about what a full turn would be. A full turn measures 360 degrees. One way to solve this is setting up a ratio of the part to the whole. We have a 45-degree angle, and a whole turn is 360 degrees. If we do some simplification here, if we divide the numerator by 45, we get one. And if we divide the denominator by 45, we get eight. That means we’re saying a 45-degree angle is one-eighth of a whole turn.

If we wanted to visualize that, we could use a circle to represent a full rotation divided in half and half again. Now we have fourths. We have quarter turns. If we divide that in half again, we’ll have eighths. A 45-degree turn is one-eighth. The question is asking, how many of those angles would it take to make a full turn? You would have to do that 45-degree angle eight times to make it back to where you started. And so we can say that it takes eight 45-degree angles to make a full turn.

In our next example, we aren’t given an image, but we’re given an arc length and a circumference. And we need to find the central angle.

A circle has a circumference of 16𝜋 units. Find, in degrees, the measure of the central angle of an arc with a length of three 𝜋 units.

To solve this question, we need to think about the relationship between the arc length and circumference and the central angle of that arc. If we take the ratio of the arc length over the circumference, it will be equal to the relationship between the central angle and a full rotation. The full rotation will either be 360 degrees or two 𝜋, depending on whether we’re working in degrees or radians.

In this case, we want to find the angle in degrees. So we’ll substitute 360 degrees for our full rotation. The arc length is three 𝜋 units, and the circumference is 16𝜋 units. If we let our central angle be 𝜃, we have 𝜃 over 360 is equal to three 𝜋 over 16𝜋.

We can simplify a little bit. The 𝜋 in the numerator and the 𝜋 in the denominator cancel out. We can’t reduce three over 16 any further. So we multiply both sides of the equation by 360 degrees. And we’ll have 𝜃 is equal to three times 360 degrees divided by 16. When we do that, we get 67.5 degrees.

In our final example, we’ll find a central angle when we’re not given the arc length or the circumference.

Find the measure of angle 𝐴𝑀𝐵.

When we look at the angle 𝐴𝑀𝐵, we see that it is a central angle of the circle. But we’re not given any information about arc 𝐴𝐵. And that means we’ll have to consider some other properties of circles and triangles to help us solve for this missing angle. The first thing we could say is that line segment 𝐴𝑀 and line segment 𝐵𝑀 are radii of the circle 𝑀. Which means we could say that line segment 𝐴𝑀 and line segment 𝐵𝑀 are equal in length to one another. Which means we can say something about triangle 𝐴𝑀𝐵. It has two equal side lengths, which makes it an isosceles triangle.

And in an isosceles triangle, the angles opposite the equal side lengths are equal in measure. So we can say the measure of angle 𝑀𝐵𝐴 will be equal to the measure of angle 𝑀𝐴𝐵. The measure of angle 𝑀𝐴𝐵 is 27 degrees. And that means we can say that the measure of 𝑀𝐵𝐴 is 27 degrees. And because triangle 𝐴𝑀𝐵 is an isosceles triangle and the sum of all the interior angles in a triangle equals 180 degrees, we can say 27 plus 27 plus the measure of angle 𝐴𝑀𝐵 is 180 degrees. If we add 27 plus 27, we get 54. And when we subtract 54 degrees from both sides of the equation, we see that the measure of angle 𝐴𝑀𝐵 is 126 degrees. The central angle 𝐴𝑀𝐵 is 126 degrees.

Now we’re ready to summarize some key points. When we have a central angle subtended by a certain arc, the ratio of the central angle to the full rotation will be equal to the arc length over the circumference of the circle. The central angle is created by two points on the outside of the circle and the center of the circle. The arc length is the curved distance between those two points. And the circumference is the distance all the way around the circle.

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