Lesson Explainer: Central Angles and Arcs Mathematics

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

We begin by defining exactly what is meant by an arc of a circle.

Definition: Arc of a Circle

An arc of a circle is a section of the circumference of the circle between two radii.

We can see some examples of arcs of circles in the following diagrams.

To help us differentiate between different arcs, we introduce the idea of the central angle.

Definition: Central Angle

A central angle of a circle is an angle between two radii with the vertex at the center. In the following diagram, ∠𝐴𝐡𝐢 is an example of a central angle.

We can extend this idea to say that the central angle of an arc is the central angle subtended by the arc.

For example, the central angles of the two given arcs are shown in the following diagrams.

We can see that the larger the central angle, the larger the arc. Therefore, it would be useful to talk about the measure of the central angle of the arc with regards to the length of the arc. We do this by introducing the following definition.

Definition: Measure of an Arc

The measure of an arc is the measure of its central angle.

For example, in the diagram below, the measure of the arc in red is 26∘.

We can notice something interesting in this diagram: there are two possible arcs from 𝐴 to 𝐡, the shorter one in red and the longer one in green. To help us differentiate between these two cases, we call the longer arc the major arc and the shorter arc the minor arc.

Definition: Major and Minor Arcs of a Circle

Given two radii, we denote the longer of the two arcs between the radii as the major arc and the shorter of the arcs as the minor arc. Equivalently, the arc with the smaller central angle is the minor arc and the arc with the larger central angle is the major arc.

To help differentiate between the major and minor arcs, we denote the minor arc as 𝐴𝐡 and label the major arc using an extra point (for example, 𝐴𝐢𝐡).

We can also use the notation π‘šπ΄π΅ for the measure of the minor arc from 𝐴 to 𝐡. In this case, we can use π‘šπ΄πΆπ΅ for the measure of the major arc from 𝐴 to 𝐡.

If the two arcs are the same length, then we call these semicircular arcs. These occur when the radii form a diameter or when their central angles have equal measures.

Since the size of the central angle of an arc determines its size, we define major and minor arcs in terms of their central angles. If the central angle is greater than 180∘, then the arc is major. If the central angle is less than 180∘, then the arc is minor. If the central angle is equal to 180∘, then the arc is semicircular.

In our first example, we will determine the measure of an arc given its central angle.

Example 1: Finding the Measure of an Arc given Its Central Angle

Find π‘šπ΄π·.

Answer

We recall that the notation π‘šπ΄π· means the measure of the minor arc between 𝐴 and 𝐷 and that the measure of an arc is defined to be its central angle. We highlight this arc on the following diagram.

The central angle of an arc is the angle at the center of the circle between the two radii subtended by the arc. For the minor arc 𝐴𝐷, this is 33∘. The measure of the arc is defined to be equal to this value. Hence, π‘šπ΄π·=33.∘

Before we move on to more examples, there is one more definition we need to discuss, which is that of adjacent arcs.

Definition: Adjacent Arcs

We say that two arcs are adjacent if they share a single point in common or if they share only their endpoints in common.

In the circle above, 𝐴𝐡 and 𝐡𝐢 are adjacent since they share a single point in common. Similarly, 𝐴𝐢 and 𝐴𝐷𝐢 are adjacent since they only share both endpoints in common.

In fact, the major and minor arcs of a circle between two points will always be adjacent.

Since the measure of an arc is the measure of its central angle and adjacent arcs will have adjacent central angles, we can find the measure of adjacent arcs by adding their measures. For example, in the circle above, we have π‘šπ΄πΆ=π‘šπ΄π΅+π‘šπ΅πΆ.

Let’s see an example of identifying adjacent arcs in a circle.

Example 2: Identifying Adjacent Arcs in a Circle

Which of the following arcs are adjacent in the given circle?

  1. 𝐴𝐡 and 𝐢𝐷
  2. 𝐴𝐡 and 𝐡𝐢
  3. 𝐴𝐷 and 𝐡𝐢
  4. 𝐴𝐢 and 𝐷𝐡

Answer

We recall that two arcs are adjacent if they share a single point in common and that the notation 𝐴𝐡 means the minor (or smaller) arc from 𝐴 to 𝐡. Therefore, we can answer this question by highlighting each pair of arcs. Let’s start with 𝐴𝐡 and 𝐢𝐷.

We see that the arcs share no points in common, so they cannot be adjacent. Next, we highlight 𝐴𝐡 and 𝐡𝐢.

We can see that 𝐴𝐡 and 𝐡𝐢 only share the point 𝐡 in common; it is the end point of both arcs, so these arcs are adjacent. For due diligence, we will also check the other options.

We have 𝐴𝐷 and 𝐡𝐢.

We can see that these arcs share no points in common, so they are not adjacent.

Finally, we check 𝐴𝐢 and 𝐷𝐡.

We can see that every point on arc 𝐡𝐢 lies on both arcs, so this pair of arcs share more than one point in common. Hence, they are not adjacent.

The only pair of arcs to share a single point in common is the pair of arcs 𝐴𝐡 and 𝐡𝐢, which is option B.

In our next example, we will determine the measure of an arc using a diagram and knowledge of the ratio of two other arc measures.

Example 3: Finding an Arc’s Measure in a Circle given the Other Arcs’ Measures by Solving Linear Equations

Given that 𝐴𝐡 is a diameter in a circle of center 𝑀 and π‘šπ΄πΆβˆΆπ‘šπ·π΅=85∢67, determine π‘šπ΄πΆπ·.

Answer

We want to determine the value of π‘šπ΄πΆπ·. We recall that this is the measure of the arc from 𝐴 to 𝐢 to 𝐷, as shown in the following diagram.

We can see that this arc consists of two adjacent arcs: 𝐴𝐢 and 𝐢𝐷. We can therefore find the measure of 𝐴𝐢𝐷 by finding the sum of the measures of 𝐴𝐢 and 𝐢𝐷.

Since the measure of an arc is equal to its central angle, π‘šπΆπ·=π‘šβˆ πΆπ‘€π·. We are given π‘šβˆ πΆπ‘€π·=28∘, so we have π‘šπΆπ·=28.∘

We know that the sum of the measures of all the arcs that make up the circle will be 360∘. In particular, the sum of the measures of the arcs that make up 𝐴𝐡 will be 180∘ since 𝐴𝐡 is a diameter. This means

π‘šπ΄πΆ+π‘šπΆπ·+π‘šπ΅π·=180π‘šπ΄πΆ+28+π‘šπ΅π·=180π‘šπ΄πΆ+π‘šπ΅π·=152.∘∘∘∘(1)

We are told that π‘šπ΄πΆβˆΆπ‘šπ·π΅=85∢67.

Therefore, the quotients of each part of the ratio must be equal: π‘šπ΄πΆπ‘šπ·π΅=8567.

We can rearrange this equation to get π‘šπ·π΅=67π‘šπ΄πΆ85.

We can substitute our expression for π‘šπ·π΅ into equation (1) and simplify to get π‘šπ΄πΆ+67π‘šπ΄πΆ85=152152π‘šπ΄πΆ85=152π‘šπ΄πΆ=ο€Ό85Γ—152152οˆπ‘šπ΄πΆ=85.∘∘∘∘

Finally, π‘šπ΄πΆπ·=π‘šπ΄πΆ+π‘šπΆπ·=85+28=113.∘∘∘

Since an arc of a circle is a portion of its circumference, we can use the circumference of the circle to determine the length of the arc. We can do this by using the measure of the arc or, equivalently, its central angle.

To help us determine the length of an arc, let’s start with an example. We want to determine the length of the minor arc in the following diagram.

First, recall that a circle of radius π‘Ÿ has circumference 2πœ‹π‘Ÿ. This means the circumference of this circle is 2πœ‹π‘Ÿ.

We can see that this arc represents one-quarter of the circle, but it is good practice to see why this is the case. A full turn is an angle of 360∘, so a 90∘ angle is 90360=14 of the circle.

Hence, the arc length is one-quarter of the circumference: arclength=14(2πœ‹π‘Ÿ)=πœ‹π‘Ÿ2.

In general, if the central angle (or arc measure) is πœƒβˆ˜, then the arc length is πœƒ360(2πœ‹π‘Ÿ)∘∘. We can state this formally as follows.

Definition: Length of an Arc

If the central angle (or measure) of an arc in a circle of radius π‘Ÿ is πœƒβˆ˜, then the length of the arc, 𝑙, is given by 𝑙=πœƒ360(2πœ‹π‘Ÿ).∘∘

In our next example, we will use the formula for the length of an arc to determine the measure of an arc that gives a specific proportion of the circumference of a circle.

Example 4: Finding the Measure of the Arc That Represents a Known Part of the Circumference of a Circle

Find the measure of the arc that represents 16 of the circumference of a circle.

Answer

To answer this question, we first recall that the length, 𝑙, of an arc of measure π‘₯∘ in a circle of radius π‘Ÿ is given by 𝑙=π‘₯360(2πœ‹π‘Ÿ).∘∘

We want this value to be equal to 16 the circumference of the circle, and we know a circle of radius π‘Ÿ has circumference 2πœ‹π‘Ÿ. So, we want 𝑙=16(2πœ‹π‘Ÿ)=13(πœ‹π‘Ÿ).

Setting these two expressions for 𝑙 to be equal gives us π‘₯360(2πœ‹π‘Ÿ)=13(πœ‹π‘Ÿ).∘∘

We can then solve for π‘₯∘. We divide through by πœ‹π‘Ÿ to get π‘₯360(2)=13.∘∘

Finally, we multiply through by 180∘ and simplify: π‘₯=13(180)=60.∘∘∘

It is worth noting there is another method of answering this question. We can note that the proportion of the measure of an arc to 360∘ is exactly the same as the proportion of the arc length to the circumference. In other words, π‘₯360=.∘∘arclengthcircumference

We are told arclengthcircumference=16, so we have π‘₯360=16,∘∘ which we solve and get π‘₯=60∘∘.

There is an important corollary to the arc length formula involving congruent arcs in a circle, which we will discuss now.

Property: Congruent Arcs

Since the length of an arc is determined by its central angle (or measure) and the radius of the circle, we can conclude that if two arcs in circles with equal radii have the same length, then their central angles (and measures) will be equal. In other words, two arcs are congruent if and only if their central angles (or measures) are equal.

For example, we can use this formula to determine the arc length in the following diagram.

The radius of this circle is 2 and the central angle is 30∘, so we have 𝑙=30360(2πœ‹(2))=πœ‹3.∘∘lengthunits

This also tells us that any arc of length πœ‹3 in this circle, or in any circle with a radius of 2 length units, will have a measure of 30∘.

Let’s now see an example of how we can apply this congruence property of arcs to determine an arc length in a circle.

Example 5: Understanding the Relationship between Arcs with Equal Lengths

Consider a circle 𝑀 with two arcs, 𝐴𝐡 and 𝐢𝐷, that have equal measures. 𝐴𝐡 has a length of 5 cm. What is the length of 𝐢𝐷?

Answer

We are told that 𝐴𝐡 and 𝐢𝐷 have the same measure and we can recall that if two arcs have the same measure, then they are congruent. So, their lengths are equal. Hence, 𝐢𝐷 has a length of 5 cm.

Although not necessary in answering this question, it can be worth seeing why this result holds true from the formula for the length of an arc. We recall that the length, 𝑙, of an arc between 𝑃 and 𝑄 in a circle of radius π‘Ÿ is given by the formula 𝑙=π‘šπ‘ƒπ‘„360(2πœ‹π‘Ÿ).

So, the length of 𝐴𝐡 is given by 𝑙=π‘šπ΄π΅360(2πœ‹π‘Ÿ).

Since π‘šπ΄π΅=π‘šπΆπ·, we have 𝑙=π‘šπ΄π΅360(2πœ‹π‘Ÿ)=π‘šπΆπ·360(2πœ‹π‘Ÿ).

However, this expression is the length of 𝐢𝐷, so their lengths are equal.

Hence, 𝐢𝐷 has a length of 5 cm.

Another similar property to the one in the example above is that if the lengths of the chords between the endpoints of two arcs in a circle are equal, then the arcs have equal measures. In fact, the same is true in reverse; if the arcs have the same measure, then the chords between their respective endpoints will have equal lengths.

To see why this is true, consider the following circle.

Let’s suppose that 𝐴𝐡 and 𝐢𝐷 have equal measures. Then, the measures of the central angles are equal: π‘šβˆ π·π‘€πΆ=π‘šβˆ π΄π‘€π΅.

We also know that 𝐴𝑀, 𝐡𝑀, 𝐢𝑀, and 𝐷𝑀 are radii, so they have the same length. Hence, triangles 𝐴𝑀𝐡 and 𝐷𝑀𝐢 are congruent by the SAS rule, so 𝐴𝐡 and 𝐷𝐢 must have the same length.

Similarly, if 𝐴𝐡 and 𝐢𝐷 have the same length, then, by using the radii of the circle, we have that triangle 𝐴𝑀𝐡 and 𝐷𝑀𝐢 are congruent by the SSS rule. So, the measures of the internal angles are equal. In particular, π‘šβˆ π·π‘€πΆ=π‘šβˆ π΄π‘€π΅.

Then, since the measures of the central angles are equal, we know that their measures (and arc lengths) are equal.

Property: Congruent Chords of Congruent Arcs

In the same circle or in congruent circles, if two arcs have the same measure, then the chords between their respective endpoints will have equal lengths. In fact, the same is true in reverse; in the same circle or in congruent circles, if two chords between the endpoints of two arcs are congruent, then the two arcs have the same measure. We can see this in the following diagram.

  1. If π‘šπ΄π΅=π‘šπ·πΆ, then 𝐴𝐡=𝐷𝐢.
  2. If 𝐴𝐡=𝐷𝐢, then π‘šπ΄π΅=π‘šπ·πΆ.

Let’s see an example of how we can apply this property.

Example 6: Understanding the Relationship between Arcs and Chords

Consider circle 𝑀 with two chords of equal lengths, 𝐴𝐷 and 𝐡𝐢. If 𝐴𝐷 has a length of 5 cm, what is the length of 𝐡𝐢?

Answer

We see that 𝐴𝐷 and 𝐡𝐢 are the chords between the endpoints of arcs 𝐴𝐷 and 𝐡𝐢 as shown.

We then recall that if the lengths of the chords between the endpoints of two arcs in a circle are equal, then the arcs have equal lengths and measures. Therefore, since 𝐴𝐷 and 𝐡𝐢 have the same length, 𝐴𝐷 and 𝐡𝐢 will also have the same length.

Hence, since 𝐴𝐷 has a length of 5 cm, 𝐡𝐢 also has a length of 5 cm.

In our next example, we will use a diagram and the properties of central angles to determine the measure of a given arc.

Example 7: Finding the Measure of an Arc in a Circle given a Diameter and the Measures of Two Central Angles in the Form of Algebraic Expressions

Given that 𝐴𝐡 is a diameter in circle 𝑀 and π‘šβˆ π·π‘€π΅=(5π‘₯+12)∘, determine π‘šπ΄πΆ.

Answer

We are asked to find π‘šπ΄πΆ, which is the measure of the minor arc from 𝐴 to 𝐢, the arc shown in the following diagram.

We recall that the measure of an arc is equal to its central angle, and we can see in the diagram the central angle for this arc is 4π‘₯∘. So, π‘šπ΄πΆ=4π‘₯∘. Therefore, we need to determine the value of π‘₯. To find the value of π‘₯, we will start by adding the angle we are given in the question to the diagram.

We then note that 𝐴𝐡 is a diameter of the circle, which means it is a straight line. So, we must have π‘šβˆ π·π‘€π΅+π‘šβˆ π·π‘€π΄=180(5π‘₯+12)+2π‘₯=180.∘∘∘∘

We can then solve this for π‘₯7π‘₯+12=1807π‘₯=168π‘₯=24.∘∘∘∘∘∘∘

Finally, we know that π‘šπ΄πΆ=4π‘₯.∘

Substituting in the value for π‘₯, we get that π‘šπ΄πΆ=4(24)=96.∘∘

Let’s finish by recapping some of the important points of this explainer.

Key Points

  • An arc of a circle is a section of the circumference of the circle between two radii.
  • A central angle of a circle is an angle between two radii with the vertex at the center.
  • The central angle of an arc is the central angle subtended by the arc.
  • The measure of an arc is the measure of its central angle.
  • Given two radii, we denote the longer of the two arcs between the radii as the major arc and the shorter of the arcs as the minor arc. Arcs of equal lengths are called semicircular arcs; these occur when the radii form a diameter.
  • An arc is major if its measure (or the measure of its central angle) is greater than 180∘, an arc is minor if its measure (or the measure of its central angle) is less than 180∘, and an arc is semicircular if its measure (or the measure of its central angle) is equal to 180∘.
  • We denote the minor arc from 𝐴 to 𝐡 as 𝐴𝐡, and major arcs can be labeled by using an extra point (for example, 𝐴𝐢𝐡).
  • We use the notation π‘šπ΄π΅ for the measure of the minor arc from 𝐴 to 𝐡 and π‘šπ΄πΆπ΅ for the measure of the major arc from 𝐴 to 𝐡 that passes through 𝐢.
  • We say that two arcs are adjacent if they share a single point in common or if they share only their endpoints in common.
  • If the central angle (or measure) of an arc in a circle of radius π‘Ÿ is πœƒβˆ˜, then its length, 𝑙, is given by 𝑙=πœƒ360(2πœ‹π‘Ÿ).∘∘
  • If two arcs in the same circle have equal lengths, then their central angles and measures are equal. The same is true in reverse; if two arcs have equal lengths, then their central angles and measures are equal.
  • If two arcs in the same circle have equal measures, then the chords between their respective endpoints have the same length. The same is true in reverse; if the chords have the same length, then the arcs between the endpoints of the chords must have equal measures.

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