In this explainer, we will learn how to find the arc length and the perimeter of a circular sector and solve problems including real-life situations.
We can begin by recalling the terminology used to describe parts of a circle. Firstly, we remember that the arc of a circle is a section of the circle between two radii. However, given two radii, there are two possible arcs between the two radii. We can see an example of this in the following diagram.
Both arcs are a section of the circle between two given radii, so to avoid confusion, we denote the larger arc as the major arc and the smaller one as the minor arc.
This is equivalent to saying that if the central angle is less than , or radians, then we know it is minor. If it is larger than these values, then it is a major arc. We can then define circular arcs as below.
Definition: An Arc of a Circle
An arc of a circle is a section of the circumference of the circle between two radii.
Given two radii, we denote the larger of the arcs as the major arc and the smaller of the arcs as the minor arc. The larger arc is the one with the largest central angle.
If the two arcs are the same length, then we call these semicircular arcs. These occur when the central angle is or radians, or equivalent, when the radii form a diameter.
We can now see how we find the length of an arc of a circle. Let’s say we have the arc below.
We can find the length of any arc subtended by an angle by first recalling how we find the circumference of a circle, the distance around the outside of the circle.
The circumference, , of a circle of radius is given by
The length of the minor arc above can be calculated by multiplying the circumference, , by . In general, for an arc with central angle , this is a section of the circumference and its length is calculated as
We can do the same for an angle measured in radians. If the central angle is radians, then the arc is a section of the circumference. Thus, the length of the arc is given by
This gives us the following formulas for finding the length of an arc of a circle.
Definition: Arc Length
The length of an arc that subtends an angle , measured in degrees, in a circle of radius is given by
The length of an arc that subtends an angle , measured in radians, in a circle of radius is given by
We will now see some examples of how we can apply these formulas, beginning with how we can find an arc length given an angle in radians.
Example 1: Calculating the Length of an Arc
Find the length of the blue arc given the radius of the circle is 8 cm, and the angle measure shown is in radians. Give the answer to one decimal place.
Answer
In this problem, we are given the angle subtended by an arc whose measure is in radians. We recall that the length of an arc that subtends an angle , measured in radians, in a circle of radius is given by
We are given that the radius of this circle is 8 cm. Thus, we can substitute and into the formula to give
As we are asked for an answer to the nearest decimal, we can use a calculator to find a decimal equivalent and approximate, which gives
Therefore, to one decimal place, the length of the blue arc is 33.5 cm.
In the next example, we will find the length of an arc in a real-world context.
Example 2: Solving an Applied Problem Involving the Arc Length of a Pendulum
A pendulum of length 26 cm swings . Find the length of the circular pathway that the pendulum makes giving the answer in centimetres in terms of .
Answer
In this question, we are given that the pendulum follows a circular path. This means that we can model the path of the pendulum as an arc of a circle. The pendulum pivots about a single point; thus, the length of the pendulum will be the radius of the circle. We are given the central angle of the arc as .
The arc length of a circle of radius with a central angle , measured in degrees, is given by
So, substituting the values and and simplifying, we have
We can leave our answer in terms of to give the length of the circular pathway as cm.
As an alternative method, we can convert the angle in degrees to one in radians and then use the formula to find the length of an arc subtending an angle in radians. We recall that to change any angle in degrees to one in radians, we multiply the angle by . Hence,
The length of an arc that subtends an angle , measured in radians, in a circle of radius is given by
Since the pendulum forms a circle of radius , we can substitute this into the formula, giving
Either method will allow us to calculate the length of the circular pathway as cm.
We can expand the process of finding the length of an arc of a circle to finding the perimeter of a circular sector. A sector of a circle is a part of the circle enclosed by two radii and an arc between them. We can recall that the perimeter of a shape is the distance around the outside edge.
The perimeter of a sector is the sum of the two radii and the arc length. We can define this below.
Definition: Perimeter of a Sector
The perimeter of the sector of a circle of radius subtended by an angle of measured in degrees is
The perimeter of the sector of a circle of radius subtended by an angle of measured in radians is
In the next example, we will see how we can find the perimeter of a circular sector by first finding the length of the arc.
Example 3: Finding the Perimeter of a Sector
The radius of a circle is 7 cm and the central angle of a sector is . Find the perimeter of the sector to the nearest centimetre.
Answer
We can sketch this circular sector in the following way.
The perimeter of the sector, the distance around the outside edge, is the sum of the lengths of the two radii along with the arc that makes the sector:
We are given that the length of the radius is 7 cm, but we will need to calculate the length of the arc.
The arc length of a circle of radius with a central angle , measured in degrees, is given by
We are given that and the central angle . Hence, substituting these into the formula above gives
We can keep this value in terms of for the next part of the calculation.
To find the perimeter, we substitute the radius and arc length into the perimeter calculation:
Thus, we have
We can then approximate this value to the nearest centimetre to give the result that the perimeter of the sector is 19 cm.
In the next example, we will use information about the perimeter of a sector to find its radius.
Example 4: Finding the Radius of a Circular Sector When Given the Central Angle and the Perimeter of the Sector
The perimeter of a circular sector is 67 cm and the central angle is 0.31 rad. Find the radius of the sector giving the answer to the nearest centimetre.
Answer
The perimeter of a sector is the distance around its outside edge. It is the length of two radii along with the arc that makes the sector. We can define the arc length as and write that
We are given that the perimeter is 67 cm, so we have the equation
We can use the information about the central angle of the sector to help us calculate the arc length, , noting that the angle measure is in radians. We recall that the length of an arc, , that subtends an angle , measured in radians, in a circle of radius is given by
We now substitute the given angle, , into this equation to find as
Next, substituting into the equation gives
Finally, rounding to the nearest centimetre, we can give the answer that the radius of the sector is 29 cm.
In the final example, we will see how we can use information about tangents intersecting to find the length of an arc.
Example 5: Finding the Length of an Arc given Two Intersecting Tangents and Their Angle of Intersection
If and the radius of the circle equals 3 cm, find the length of the major arc .
Answer
The major arc will be the larger of the two arc lengths, as shown in the following diagram.
In order to find the length of either the major or the minor arc , we will need to establish the measure of the central angle subtending the arc. We can sketch the radii from and to the center, , of the circle.
We recall that a tangent to the circle at a point meets the radius of the circle from at , so and . We can add this information to the diagram, along with the given information that .
We observe that we now have a quadrilateral, , and three of the angle measures within it. The sum of the internal angles in a quadrilateral is ; thus,
Substituting in the values for the angles and simplifying, we have
We can now use this information, that , to help us find the length of the major arc .
The arc length of a circle of radius with a central angle , measured in degrees, is given by
Here, if we use , then this will give us the length of the minor arc . There are two alternative options to find the length of the major arc. In the first method, we find the reflex by calculating . Substituting and into the formula gives
We can keep this value in terms of , or alternatively we can find the decimal equivalent as cm and round to one decimal place to give the answer for the length of the major arc as 13.4 cm.
In the second, alternative method, when we initially calculated that , we could find the length of the minor arc by substituting , along with the radius, , to give the length of the minor arc as
If we have the length of either arc and we wish to find the other, we can recognize that the sum of the two arcs is equal to the circumference of the circle. For a circle of radius , the circumference, , is given by
To find the circumference, we substitute the radius into the formula, giving
Now, to find the length of the major arc , we can calculate
Either method has demonstrated that the length of the major arc , rounded to one decimal place, is 13.4 cm.
We now summarize the key points.
Key Points
- An arc of a circle is a section of the circumference of the circle between two radii.
- The larger of the two arcs is the major arc and the smaller is the minor arc. If the central angle between the two radii is less than , or radians, then it is a minor arc. If it is larger than these values, then it is a major arc. If the angle is exactly , or radians, then there will be two semicircular arcs.
- The length of an arc that subtends an angle , measured in degrees, in a circle of radius is given by
- The length of an arc that subtends an angle , measured in radians, in a circle of radius is given by
- The perimeter of a sector is the sum of the lengths of two radii along with the arc that makes the sector.
