Lesson Video: Central Angles and Arcs Mathematics

Video Transcript

Central Angles and Arcs

In this video, we will learn how to identify central angles, use their measures to find the measures of arcs, identify adjacent arcs, find arc lengths, and identify congruent arcs in congruent circles.

Let’s start by defining exactly what we mean by an arc of a circle. An arc of a circle is a section of the circumference of the circle between two radii. For example, consider the following circle centered at the point 𝑀 with two radii included. The section marked in orange is an example of an arc of a circle. It’s a section of the circumference of the circle between two radii. And it’s worth noting the exact same thing is true for the part marked in blue. It’s also a section of the circumference of the circle between two radii. Therefore, if we’re given two radii of a circle, we can’t just say the arc of a circle. We need to differentiate between these two cases. We call the longer of the two arcs the major arc. And we call the shorter of the two arcs the minor arc.

There is one more possibility. The two arcs could have the same length. In this case, they make up half the circle each. So we call these semicircular arcs. And these will only occur when our two radii form a diameter of the circle.

Now that we’re familiar with the concept of arcs, let’s discuss how we denote arcs in a circle. First, because arcs are defined between two radii of a circle, we can also talk about the points of intersection with the radii and the circle. For example, we can talk about the points 𝐴 and 𝐵 in this diagram. Then, we could say that the arc marked in orange is the minor arc from 𝐴 to 𝐵 and the arc marked in blue is the major arc from 𝐴 to 𝐵. We denote the arc from 𝐴 to 𝐵 by using the following notation. And unless otherwise specified, this means the minor arc.

Finally, it will be useful to introduce a concept of adjacent arcs. We say that two arcs are adjacent arcs if they only share a single point in common or they share both endpoints in common, where the endpoints of an arc are the points where the arcs end. For example, in our diagram, both the major and minor arc have endpoints 𝐴 and 𝐵. And since both of these arcs share both of their endpoints in common, the major and minor arc between 𝐴 and 𝐵 are adjacent. But this is not the only possible way to have adjacent arcs. We could introduce another point 𝐶 not on the minor arc from 𝐴 to 𝐵 as shown. Then, we can see in our diagram the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐵 to 𝐶 share a single point in common, the point 𝐵. So the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐵 to 𝐶 are adjacent. Let’s now see an example where we’re asked to identify the adjacent arcs in a circle.

For the given circle, which of the following arcs are adjacent? Option (A) the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐶 to 𝐷. Option (B) the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐵 to 𝐶. Option (C) the minor arc from 𝐴 to 𝐷 and the minor arc from 𝐵 to 𝐶. Or option (D) the minor arc from 𝐴 to 𝐶 and the minor arc from 𝐷 to 𝐵.

In this question, we’re given a circle and we need to determine which of the given pairs of arcs in this circle are adjacent. To answer this question, let’s start by recalling what it means for two arcs in our circle to be adjacent. We say that two arcs are adjacent if they share only a single point or only both endpoints. Therefore, we can determine which pair of arcs are adjacent by sketching them and determining how many points they share in common. Let’s start with option (A). We’ll sketch the minor arc from 𝐴 to 𝐵. Remember, there’s two arcs in our circle from 𝐴 to 𝐵, and we want the shorter one. Next, we’ll sketch the minor arc from 𝐶 to 𝐷. Once again, this is the shorter section of the circumference of our circle from 𝐶 to 𝐷. And we can see that these two arcs share no points in common. So they’re not adjacent.

We can do the same for option (B). Let’s sketch the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐵 to 𝐶. The minor arc from 𝐴 to 𝐵 is the shorter section of the circumference of our circle between 𝐴 and 𝐵. And the minor arc from 𝐵 to 𝐶 is the shorter section of the circumference of our circle between 𝐵 and 𝐶. We can see in our sketch both of these arcs contain the point 𝐵. In fact, it’s the only point they share in common. And this means that they’re adjacent. So the answer to this question is option (B). We could stop here. However, for due diligence, let’s check the other two options.

To check option (C), we need to sketch the minor arc from 𝐴 to 𝐷 and the minor arc from 𝐵 to 𝐶. If we do this, we get the following. We can see that these two arcs share no points in common, so they’re not adjacent. Finally, let’s look at option (D). We need to determine whether the minor arc from 𝐴 to 𝐶 and the minor arc from 𝐷 to 𝐵 are adjacent. We can sketch both of these arcs onto our circle, and we notice something interesting. Every single point between 𝐵 and 𝐶 on our circle lies in both arcs. So although they do share a point in common, they actually share an infinite number of points in common. So these arcs are not adjacent. Therefore, of the given options, only the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐵 to 𝐶 are adjacent, which was option (B).

We’re now almost ready to determine the length of an arc. However, there’s a few more definitions we need first. We start by noting that an arc is a portion of the circumference of a circle. Therefore, if we knew this proportion, we would be able to use this to determine the arc length. It would be the circumference multiplied by the proportion. And to determine this proportion, we need to introduce the concept of a central angle.

The central angle of an arc between two radii is the angle at the center of the circle between the two radii subtended by the arc. So, in our diagram, the angle 𝜃 is the central angle for the minor arc 𝐴𝐵. Similarly, 360 degrees minus 𝜃 will be the central angle for the major arc 𝐴𝐵. And we can see the larger the central angle, the larger the arc of the circle will be. In fact, this allows us to determine major or minor arcs by their central angle. If the central angle of an arc is between zero and 180 degrees, the arc will be a minor arc. If the central angle is equal to 180 degrees, we will have a semicircular arc. And if it’s bigger than 180 degrees, we will have a major arc.

Before we move on to arc length, there’s one more thing we need to define, the measure of an arc. The measure of the minor arc from 𝐴 to 𝐵, written measure of minor arc 𝐴 to 𝐵, is equal to its central angle. For example, if the central angle of the minor arc from 𝐴 to 𝐵 was 60 degrees, then we could say the measure of this arc is 60 degrees.

We’re now ready to determine the arc length. Let’s start with an example. Let’s say we have a circle of radius 𝑟 and an arc with central angle 90 degrees. First, we recall the circumference of our circle is two 𝜋 multiplied by the radius. The circumference is two 𝜋𝑟. We want to determine the length of the arc which has central angle 90 degrees. If we add the points 𝐴 and 𝐵, this is the minor arc from 𝐴 to 𝐵.

We can see in our diagram that this makes up one-quarter of the circle. So we could just multiply the circumference by one-quarter to find the length of this arc. However, it’s good practice to remember exactly why this makes up one-quarter of the circle. A full turn around the circle is 360 degrees, and our central angle is 90 degrees. 90 degrees divided by 360 degrees is one-quarter. This confirms the length of this arc will be one-quarter of the circumference of the circle, one-quarter times two 𝜋𝑟, which we can of course simplify to 𝜋𝑟 divided by two.

In general though, the central angle or measure of our arc will not be 90 degrees. Instead, it will be some angle 𝜃 degrees. However, we can find the length of the arc in exactly the same way. The proportion of the circumference of the circle represented by the arc will be 𝜃 degrees divided by 360 degrees. So the length of the arc will be the circumference of the circle multiplied by this proportion. It will be 𝜃 degrees divided by 360 degrees multiplied by two 𝜋𝑟. We can write these formulae as follows. If the central angle or measure of an arc in a circle of radius 𝑟 is 𝜃 degrees, then the length of the arc 𝐿 is given by the following formula. 𝐿 is equal to 𝜃 degrees divided by 360 degrees multiplied by two 𝜋𝑟. Let’s see an example of applying this formula.

Let’s say we have a circle of radius two and an arc of measure 30 degrees. Then, the length of the arc 𝐿 is equal to the measure of the arc 30 degrees divided by 60 degrees multiplied by two 𝜋 times the radius, which is two. And if we simplify this expression, we see it’s equal to 𝜋 by three. And remember, this represents a length, so we can say this has length units.

Let’s now see an example of applying some of these definitions to a question.

Find the measure of the minor arc from 𝐴 to 𝐷.

In this question, we’re asked to determine the measure of the minor arc from 𝐴 to 𝐷. Let’s start by sketching this arc onto our diagram. Remember, the minor arc from 𝐴 to 𝐷 is the shorter section of the circle between 𝐴 and 𝐷. And we were asked to find the measure of this arc. And we recall the measure of an arc is equal to the measure of its central angle. And the central angle of an arc is the angle at the center of a circle which is subtended by the arc. In this case, the central angle of the minor arc from 𝐴 to 𝐷 is the angle 𝐷𝑀𝐴. And we’re told that the measure of this angle is 33 degrees. And the measure of the minor arc from 𝐴 to 𝐷 must be equal to this value. Therefore, the measure of the minor arc from 𝐴 to 𝐷 is 33 degrees.

Let’s now see an example involving two arcs with the same measure.

Given circle 𝑀 with two arcs from 𝐴 to 𝐵 and 𝐶 to 𝐷 that have equal measures and that the arc from 𝐴 to 𝐵 has a length of five centimeters, what is the length of the arc from 𝐶 to 𝐷?

In this question, we’re given a circle. And we’re told that two of its arcs have the same lengths, the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐶 to 𝐷.

We can add both of these to our diagram. Remember, an arc of a circle is a section of the circumference of a circle, and the minor arc will be the shorter arc between the two points. And we’re told that the minor arc from 𝐴 to 𝐵 has a length of five centimeters, so we can also add this to our diagram. We need to use this to determine the length of the arc from 𝐶 to 𝐷. To answer this question, let’s start by recalling what the measure of an arc means. The measure of an arc is the measure of its central angle. That’s the angle at the center of the circle, which is subtended by the arc. For example, the angle 𝐴𝑀𝐵 is the central angle of the minor arc from 𝐴 to 𝐵. And the angle 𝐷𝑀𝐶 is the central angle of the minor arc 𝐶𝐷. And since the measure of these two arcs are equal, the measures of their central angles must also be equal.

Let’s then say that these angles have a measure of 𝜃 degrees. Now, we can determine an expression for the lengths of both of these arcs. First, we recall the following formula for finding the length of an arc 𝐿. If its central angle is 𝜃 degrees and the radius of the circle is 𝑟, then 𝐿 is equal to 𝜃 degrees divided by 360 degrees multiplied by two 𝜋𝑟. We can apply this formula to the arc 𝐴𝐵. We know its length is five centimeters; its central angular measure is 𝜃 degrees. However, we don’t know the radius of this circle. We’ll just call this value 𝑟. We get five is 𝜃 degrees divided by 360 degrees multiplied by two 𝜋𝑟.

We can do the same for the length of the minor arc from 𝐶 to 𝐷. We’ll call this value 𝐿. The central angle of this arc is also 𝜃 degrees. So we get 𝐿 is 𝜃 degrees divided by 360 degrees multiplied by two 𝜋𝑟. We can then see the right-hand side of both of these equations are equal. Therefore, the left-hand sides must also be equal. Therefore, the length of the minor arc from 𝐶 to 𝐷 is five centimeters. And in fact, this result is true in general. If two arcs in congruent circles have the same measure, then their lengths are equal. And the reverse result is also true. If two arcs in congruent circles have the same length, then their measures must also be equal. But in this question, we were able to show the length of the arc from 𝐶 to 𝐷 is five centimeters.

We can show a very similar result to the one in the previous question. We want to show if the chords between two points 𝐴𝐵 and 𝐶𝐷 have equal length, then the arc lengths between these points also have equal length. In fact, the exact same result will be true in reverse. If two arcs 𝐴𝐵 and 𝐶𝐷 have equal length, then the chords between them will also have equal length. We’ll prove these results in both directions.

Let’s start by assuming the arc from 𝐴 to 𝐵 and the arc from 𝐶 to 𝐷 have equal length. By using the results in the previous question, since these arcs have equal lengths, their central angles must be equal. We also know that 𝐴𝑀, 𝐵𝑀, 𝐶𝑀, and 𝐷𝑀 are radii. They all have the same length. We can now see that triangles 𝐴𝑀𝐵 and 𝐶𝑀𝐷 are congruent by the side-angle-side criterion. Therefore, the side 𝐴𝐵 and the side 𝐶𝐷 must have equal length. Therefore, if the arcs have equal lengths then the chords between them have equal lengths, now let’s start by assuming that the chords have equal lengths. The radii of the circle still all have the same length. And once again, we see triangles 𝐴𝑀𝐵 and 𝐶𝑀𝐷 are congruent, this time by the side-side-side rule. And congruent triangles have the same angle. So the measure of angle 𝐴𝑀𝐵 and the measure of angle 𝐶𝑀𝐷 are equal. And therefore, the arc length of 𝐴𝐵 is equal to the arc length 𝐶𝐷 because their central angles are the same. Let’s now see an example of how we can apply this property.

Given a circle 𝑀 with two chords 𝐴𝐷 and 𝐵𝐶 that have equal lengths and the arc from 𝐴 to 𝐷 with a length of five centimeters, what is the length of the arc from 𝐵 to 𝐶?

We’re given two chords in a circle with equal lengths, 𝐴𝐷 and 𝐵𝐶. We can highlight these chords on our diagram and the fact that they have equal length. We’re also told that the length of the minor arc from 𝐴 to 𝐷 is five centimeters. We can also add this to our diagram. We need to use this to determine the length of the minor arc from 𝐵 to 𝐶. We can answer this question geometrically by noticing 𝐴𝑀, 𝐵𝑀, 𝐶𝑀, and 𝐷𝑀 are radii of the circle, so they have equal length. This means that triangles 𝐴𝑀𝐷 and 𝐵𝑀𝐶 are congruent. So the measures of the central angles of these two arcs are equal.

Then, since the central angles of these two arcs are equal, the lengths are equal, meaning that 𝐵𝐶 has length five centimeters. However, we could’ve also answered this question by just recalling if the chords between two points on a circle are equal, then their arc lengths are also equal. Using either method, we were able to show the length of the minor arc from 𝐵 to 𝐶 is five centimeters.

Let’s now go through one final example.

Given that the line segment 𝐴𝐵 is a diameter in circle 𝑀 and the measure of angle 𝐷𝑀𝐵 is five 𝑥 plus 12 degrees, determine the measure of the minor arc from 𝐴 to 𝐶.

In this question, we’re given information about a circle. We need to use this information to determine the measure of the minor arc from 𝐴 to 𝐶. Let’s start by adding the minor arc from 𝐴 to 𝐶 onto our diagram. We can recall that an arc of a circle between two points is the proportion of the circumference of the circle between those two points. And unless stated otherwise, this means the minor arc, which is the shorter of the two arcs. So the minor arc from 𝐴 to 𝐶 is shown on our diagram. We’re also told that the measure of angle 𝐷𝑀𝐵 is five 𝑥 plus 12 degrees. So we can also add this value to our diagram.

We want to determine the measure of arc 𝐴𝐶. And we recall the measure of an arc is equal to the measure of its central angle. And the central angle of an arc is the angle at the center of a circle subtended by the arc. And we’re given the measure of this angle in the diagram. The measure of arc 𝐴𝐶 is four 𝑥 degrees. Therefore, to determine the measure of arc 𝐴𝐶, we need to find the value of 𝑥. To do this, we’re going to use the fact that the line segment 𝐴𝐵 is a diameter of the circle. In particular, because 𝐴𝐵 is a diameter of the circle, the angles 𝐵𝑀𝐷 and 𝐷𝑀𝐴 make up a straight line. The sum of their measures will be 180 degrees. Therefore, we have 180 degrees is equal to two 𝑥 degrees plus five 𝑥 plus 12 degrees.

We can then solve this equation for 𝑥. We start by removing the degrees symbol, and then we simplify to get 168 is equal to seven 𝑥. Dividing both sides of the equation through by seven gives us that 𝑥 is equal to 28. Finally, we can substitute this value of 𝑥 into our equation for the measure of arc 𝐴𝐶. The measure of arc 𝐴𝐶 is four times 28 degrees, which we can calculate is 96 degrees. Therefore, given that the line segment 𝐴𝐵 in circle 𝑀 was a diameter and the measure of angle 𝐷𝑀𝐵 was five 𝑥 plus 12 degrees, we were able to determine the measure of arc 𝐴𝐶 is 96 degrees.

Let’s now go over the key points of this video. First, we saw that an arc of a circle is a section of the circumference between two radii. We also saw, given two radii, there are two possible arcs between them. We call the longer of these two arcs the major arc and the shorter of these two arcs the minor arc. And instead of thinking about an arc between two radii, we can think of an arc between two points on the circumference of a circle. And the minor arc from 𝐴 to 𝐵 is denoted by the following notation. We also called the central angle of an arc the angle at the center of the circle between the two radii, which is subtended by the arc. And we also defined the measure of an arc to be the measure of its central angle. And the measure of the minor arc from 𝐴 to 𝐵 is written in the following notation.

We also proved if the central angle or measure of an arc in a circle of radius 𝑟 is 𝜃 degrees, then the length 𝐿 of the arc is given by 𝐿 is equal to 𝜃 degrees divided by 360 degrees multiplied by two 𝜋𝑟. Finally, we showed two useful results. First, if two arcs in congruent circles have the same length, then we showed that their central angles will have equal measure. And we also showed that the same result was true in reverse. If the central angles of two arcs in congruent circles have equal measure, then the two arcs will have the same length. And we showed a very similar result. If two arcs in congruent circles have the same length, then the chords between the respective endpoints of these two arcs will also have equal length.

And finally, we showed the same result was true in reverse. If two chords in congruent circles had equal length, then the arcs between the respective endpoints of those two chords will have equal length.