Lesson Video: Arc Lengths Mathematics

Video Transcript

In this video, we’ll learn how to find the arc length and the perimeter of a circular sector and solve problems including real-life situations. But let’s first begin by recalling how we describe parts of a circle.

An arc of a circle is defined as a section of the circle between two radii. So here we have a radius joining the circle at a point 𝐴 and another radius joining the circle at point 𝐵. However, as we look at the circle, we might notice that, in fact, there are two arcs. We have this smaller arc in the shorter distance between 𝐴 and 𝐵, and then we have this larger arc. Both of these arcs would be defined in the same way in that they’re both sections of the circle between two radii. We can get around this problem of definition by defining that the smaller arc is called a minor arc and the larger arc is called the major arc. If we ever have the situation where the two radii are, in fact, the diameter or the central angle is 180 degrees or 𝜋 radians, then we would say that we have two semicircular arcs.

Now we can have a think at how we would actually find the length of any of these arcs. Let’s take this example problem. We have two radii creating a section of the circle and the angle at the center here is given as 90 degrees. What we want to do is work out the length of this minor arc. In order to help us, we can remember that the circumference, that’s the distance around the outside of the circle, is calculated by two times 𝜋 times the radius. But we don’t actually want to calculate the whole way around this circle. We only want this section. And we know that this section must be one-quarter of the whole circle. So we multiply one-quarter by two times 𝜋 times the radius.

This method works for any given central angle, if instead of having a 90-degree angle we had an angle of 𝜃 degrees and so the circumference would be multiplied by this proportion 𝜃 over 360. We can define this more formally by saying that the length of an arc which subtends an angle 𝜃 measured in degrees in a circle of radius 𝑟 is given by arc length equals two 𝜋𝑟𝜃 over 360.

You might notice this terminology that the angle 𝜃 is measured in degrees because, of course, there are other ways in which we can measure angles. One of these ways is by using a measure of radians. And we can also find the arc length when the angle is given in radians. If this central angle is measured as 𝜃 radians, then by remembering that there are two 𝜋 radians in 360 degrees, that means that we multiply the circumference two 𝜋𝑟 by the proportion of 𝜃 over two 𝜋. We can simplify this calculation by taking out a factor of two 𝜋 from the numerator and denominator, which leaves us with 𝑟𝜃.

So now we have another similar definition. This time, the length of an arc when the central angle is measured in radians is given by arc length is equal to 𝑟𝜃. We can apply either of these formulas, depending on whether the angle measure is given in degrees or radians. In the first example, we’ll see how we can find an arc length when the angle is given in radians.

Find the length of the blue arc given the radius of the circle is eight centimeters, and the angle measure shown is in radians. Give the answer to one decimal place.

In this question, we need to calculate the length of this blue arc, which is the larger of the two arcs; that’s also called the major arc. We’re given that the angle measure is in radians. We can recall that the length of an arc subtending an angle 𝜃 measured in radians in a circle of radius 𝑟 is given by arc length equals 𝑟𝜃. We can then simply plug in the information that we’re given. The radius is eight centimeters, and the angle measure is four 𝜋 over three. And we multiply these together. This gives us 32𝜋 over three. And because it’s a length, the units will be in centimeters.

We could leave our answer in this form. However, this question asks us to give the answer to one decimal place. So we’ll need to use our calculators. This gives us the values 33.510 and so on centimeters, which we can round to one decimal place as 33.5 centimeters. And so, we can give the answer that the length of the blue arc is 33.5 centimeters.

In the next question, we’ll see how we can find the length of an arc in a real-world context.

A pendulum of length 26 centimeters swings 58 degrees. Find the length of the circular pathway that the pendulum makes giving the answer in centimeters in terms of 𝜋.

In this question, we’re given that there’s a pendulum, which is 26 centimeters long. This means that the length of string from the pivot point here at the top to the ball at the end is 26 centimeters. We’re told that the angle that this pendulum swings through is 58 degrees. And we’re told that it swings in a circular pathway. We could draw a smaller diagram of the pendulum, which allows us to say that if this pendulum was to swing the entire way round, it would in fact create a circle. The length of the string, which is 26 centimeters, would in fact be the radius of the circle.

The length that we need to work out is marked in green and that’s an arc of the circle. Because we’re given that the central angle is a measurement in degrees, then we use this formula that the arc length of a circle of radius 𝑟, with a central angle 𝜃 degrees, is given by arc length equals two 𝜋𝑟𝜃 over 360. We can remember that this formula is a result of multiplying the circumference, which is two 𝜋𝑟 by this proportion of 𝜃 over 360 degrees.

Now all we need to do is plug in the values that the radius is 26 centimeters and 𝜃, the central angle, is 58 degrees. If we wish, we can take out this common factor of two before we simplify to give us the answer that the arc length is 377 over 45𝜋 centimeters. In some questions, we might be asked for a decimal approximation for the length. However, this question asks us for the length in terms of 𝜋. Therefore, we leave the answer as it is. So the circular pathway has a length of 377 over 45 𝜋 centimeters.

We’ll now have a look at how we can use what we know about finding the arc length to find the perimeter of a circular sector. So far, we’ve seen that there are two alternative formulas to find the arc length of a sector. And those two formulas are dependent on whether the central angle is given in degrees or radians. But sometimes, of course, we might need to find the perimeter of a sector. And remember that that’s just the distance around the outside. Because the two extra lengths are both radii of the circle, then to find the perimeter in each case, whether it’s in degrees or radians, then we simply add on two 𝑟 to the calculation for arc length. Let’s have a look at an example of how we do this.

The radius of a circle is seven centimeters and the central angle of a sector is 40 degrees. Find the perimeter of the sector to the nearest centimeter.

Let’s begin by sketching this circular sector. In order to find the perimeter, that’s the distance around the outside of the sector, we’ll have these two straight lengths, which will both be radii of the circle, along with this length, which is the arc of the circle. To find the arc length of a circle when the central angle 𝜃 is given in degrees, we calculate two 𝜋𝑟𝜃 over 360 where 𝑟 is the radius. We can then plug in the values into this formula. The radius is seven centimeters and the central angle is 40 and the simplified answer will be 14𝜋 over nine. And because this is a length, then the units will be centimeters.

Remember that this is simply just the arc length that we’ve calculated and we still need to work out the value for the perimeter. To calculate the perimeter, we take the arc length, which we’ve kept in terms of 𝜋 to give us the most accurate answer. And then we add on two times the length of the radius, which is two times seven. When we calculate 14𝜋 over nine plus 14, we could keep the answer in terms of 𝜋, but this time we’re asked for the answer to the nearest centimeter, so we’ll need to find a decimal equivalent for the value of the perimeter, which is 18.886 and so on centimeters. Rounding this value to the nearest centimeter gives us that the perimeter of this sector is 19 centimeters.

We’ll now have a look at an example where we’re given the perimeter of a sector and we need to calculate the radius.

The perimeter of a circular sector is 67 centimeters and the central angle is 0.31 radians. Find the radius of the sector giving the answer to the nearest centimeter.

We can sketch this circular sector as shown with its central angle of 0.31 radians. We’re given that the perimeter of this sector is 67 centimeters. And we remember that the perimeter is the distance around the outside. To find the perimeter, we’d have these two straight lengths, which will be the radius of the circle which we can define as 𝑟, plus this outer edge, which will be the arc length. We can define this arc length with the letter 𝑙. To calculate the perimeter then, we would have two times the radius two 𝑟 plus 𝑙. Given the information that the perimeter is 67 centimeters, we can write the equation that 67 equals two 𝑟 plus 𝑙.

We can’t do much with this equation at the minute because we don’t know the value of 𝑟, the radius. In fact, that’s what we need to calculate. So let’s see if we can do anything with 𝑙, which is the arc length. We remember that to find the length of an arc subtending an angle 𝜃 in radians in a circle of radius 𝑟, then we calculate arc length equals 𝑟𝜃. Here, the arc length 𝑙 is equal to 𝑟𝜃 and we know that 𝜃 is 0.31 radians. We can then substitute 𝑙 is equal to 0.31𝑟 into the equation above. This gives us 67 is equal to two 𝑟 plus 0.31𝑟, which simplifies to 67 is equal to 2.31𝑟. Then to find the value of 𝑟, we divide both sides by 2.31.

We could leave our answer as a simplified fraction, but because we’re asked for the answer to the nearest centimeter, let’s find a decimal approximation. So 𝑟 is equal to 29.004 and so on, and because this is a length and we’re dealing with centimeters, then the radius 𝑟 will also be in centimeters. Rounding this to the nearest centimeter then gives us the answer that the radius of this sector is 29 centimeters.

We’ll now have a look at one final example where we use information about tangents intersecting to find the length of an arc.

If the measure of angle 𝐴 equals 76 degrees and the radius of the circle equals three centimeters, find the length of the major arc 𝐵𝐶.

Let’s start by filling in the information that we’re given. The measure of angle 𝐴 is 76 degrees and the radius of the circle is three centimeters. We remember that an arc of a circle is a section of the circumference between two radii. In fact, here we would have two arcs, which could both be called arc 𝐵𝐶. This is why, when we’re dealing with arcs, the larger arc will be referred to as the major arc and the smaller arc is called the minor arc. In this question, we’ll need to calculate the length of the major arc. In order to work out either the minor or the major arc 𝐵𝐶, we would need to establish the measure of the central angle subtending the arc.

Let’s see if we can work out this angle 𝜃 by using the information about the tangents. We can recall that a tangent to the circle at a point 𝑝 meets the radius of the circle from 𝑝 at 90 degrees. This means that we’ll have a 90-degree angle here at 𝐶 and a 90-degree angle at 𝐵. If we labeled the center of the circle with the letter 𝑂, then we might observe that we have in fact got a quadrilateral 𝐴𝐵𝑂𝐶. We know that the sum of the internal angles in a quadrilateral is 360 degrees. This means that we can write that the measure of the four angles in the quadrilateral 𝐴, 𝐵, 𝑂, and 𝐶 must add up to 360 degrees.

We can then plug in the angle measurements: 𝐴 is 76 degrees, 𝐵 is 90 degrees, and 𝐶 is also 90 degrees. Simplifying, we have that 256 degrees plus the measure of angle 𝑂 is 360 degrees. Subtracting 256 degrees from both sides then gives us that the measure of angle 𝑂 is 104 degrees. Now that we’ve found this angle 𝜃, the measure of angle 𝑂, as 104 degrees, we can calculate an arc length. Notice, however, that if we use the angle of 104 degrees, then the arc length that we calculate will be the length of the minor arc. So there are two ways to approach this problem and find the length of the major arc instead.

The first way to approach this problem is by considering what this reflex angle 𝑂 would be and using that directly to calculate the length of the major arc. Well, because the angles about a point add up to 360 degrees, then if we subtract 104 degrees from 360, we get 256 degrees. That means that the central angle in the major arc 𝐵𝐶 will be 256 degrees. To calculate the arc length of a circle of radius 𝑟 with a central angle of 𝜃 measured in degrees, then we calculate arc length equals two 𝜋𝑟𝜃 over 360. We then simply substitute in the information. The radius 𝑟 is given as three centimeters. And we know that this central angle 𝜃 is 256 degrees.

Simplifying this gives us the arc length as 64𝜋 over 15 centimeters. We can keep this answer in terms of 𝜋, or we can find the decimal equivalent as 13.404 and so on centimeters, which rounded to one decimal place would give us the length of 𝐵𝐶 as 13.4 centimeters. Let’s make a note of this answer and have a look at the alternative method to calculate the major arc. Let’s return to the point in our working where we had calculated that this obtuse angle 𝑂 is 104 degrees. Instead of immediately calculating the major arc length, let’s calculate this smaller arc, the minor arc length. The value of the radius that we plug into the formula will still be the same at three centimeters, but the central angle this time will be 104 degrees. When we calculate this and simplify, we get an answer of 26𝜋 over 15 centimeters.

So how do we go about getting from the length of the minor arc to the length of the major arc? Well, the relationship between the major and the minor arc is that if we add these together, we would get the circumference of the circle. That’s the distance around the outside edge. The circumference is calculated by two times 𝜋 times the radius. In this case, as the radius is three, we’d have two times 𝜋 times three, which is simplified to six 𝜋 centimeters.

So now, to calculate the major arc length 𝐵𝐶, we have the circumference subtract the minor arc length 𝐵𝐶. When we substitute the values six 𝜋 minus 26𝜋 over 15 and simplify, we get the value 64𝜋 over 15 centimeters. This gives us the same decimal approximation as we found before, 13.4 centimeters. Therefore, we have confirmed the answer that, to one decimal place, the length of the major arc 𝐵𝐶 is 13.4 centimeters.

We can now summarize the key points of this video. We began by defining that an arc of a circle is a section of the circumference of a circle between two radii. We saw that the larger of two arcs is the major arc and the smaller is the minor arc. We can use the size of the central angles to help us define if the arc is major or minor. Next, we saw how we can derive two formulas for the length of an arc, depending on whether the central angle 𝜃 is measured in degrees or radians. Finally, we saw that the perimeter of a sector is the sum of the length of two radii, along with the arc that makes the sector.