Lesson Video: Arc Lengths and Sectors: Degrees Mathematics

Video Transcript

In this video, we will learn how to find the arc length and the area of a sector given the angle measure in degrees.

Now, you may be aware of an alternative way of measuring angles using radians. And in this case, the methods and formulae needed are a little different. So, it’s important to remember that throughout this video, we will be working solely with angles given in degrees. Firstly, let’s remind ourselves what we mean by arcs and sectors. If we have a circle and then we draw in two radii, the portion of the circle enclosed by these two radii and part of the circle circumference is called a sector. The portion of the circumference itself is called an arc. The angle formed between the two radii, which create the sector, is called the central angle, and this will be any value up to 360 degrees.

We also recall that the measure of an arc is defined to be equal to the central angle of its sector. Unless we divide the circle exactly in half, we will always have a minor sector and a minor arc with a central angle less than 180 degrees and a major sector and major arc with a central angle greater than 180 degrees. Let’s think then about how we can calculate the length of an arc with a central angle of 𝜃 degrees. We know that to calculate the circumference of a circle, we use the formula 𝜋𝑑 or sometimes two 𝜋𝑟.

When we have an arc, though, we need to consider that this is only part of that circumference of a full circle, and so its length will be some fraction of the total. The fraction we have will be determined by the measure of the central angle. We know that there are 360 degrees in a full turn. So whatever the measure of the central angle is, which we can think of as 𝜃, we will have this angle out of 360 as the fraction of the full circle. We can, therefore, use the formula 𝜃 over 360 multiplied by 𝜋𝑑 to calculate the length of an arc with a central angle of 𝜃 degrees.

In the same way, we can find the area of a given sector. This will be part of the area of the full circle, which is given by 𝜋𝑟 squared. Again, the fraction of the circle we have will be 𝜃 out of 360. So to find the area of a sector, we can multiply the area of the full circle by this fraction, given the formula 𝜃 over 360 multiplied by 𝜋𝑟 squared. Again, we should remember at this point that these formulae are only valid if the angle 𝜃 is given in degrees because we’ve used the fact that there are 360 degrees in a full turn.

Now that we’ve identified the formulae we’ll need, let’s look at some examples.

Determine the length of arc 𝐴𝐵𝐶 to the nearest hundredth, given that the circle has radius nine.

We are asked then to calculate the length of this arc which we recall is part of the circumference of the full circle. The formula for calculating our arc length is 𝜃 over 360 multiplied by 𝜋𝑑, where 𝜃 is the central angle of the sector. What we’re doing here is calculating the circumference of the full circle and then multiplying by the fraction of the circle that we have.

We’re told that the radius of this circle is nine units, and this in turn means that the diameter of the circle, which is always twice the radius, will be 18 units. The central angle of this sector is given as 72 degrees. So, substituting 72 for 𝜃 and 18 for 𝑑, we have that the length of arc 𝐴𝐵𝐶 is 72 over 360 multiplied by 18𝜋. Now, we can actually simplify this because 72 is a factor of 360 and the fraction simplifies to one-fifth. So if we were looking to give an exact answer to this problem, we could give our answer as 18 over five multiplied by 𝜋 or eighteen-fifths of 𝜋.

However, the question asks us to give our answer to the nearest hundredth, so we’ll use our calculator to evaluate this. We get 11.30973. And then rounding to the nearest hundredth, we have 11.31. There were no units given for the radius in the questions, so there are no units for our answer. But there would be some kind of length units, such as centimeters or feet. Our answer to the problem is 11.31.

In our next example, we’ll see how we can work backwards from knowing the length of an arc to determining the arc’s measure or the central angle of the sector.

An arc on a circle with a radius of 50 has a length of 115. Determine the arc’s measure to the nearest tenth of a degree.

Let’s summarize what we know on a quick sketch of this problem. We have circle with a radius of 50 units. An arc which is part of the circumference of this circle has a length of 115 units. We’re then asked to determine the arc’s measure, which we recall is equal to the central angle of the sector. We can use the Greek letter 𝜃 to represent this angle. Now, we recall that the formula for calculating an arc length is 𝜃 over 360 multiplied by 𝜋𝑑. We multiply the circumference of the full circle by the fraction of the circle that we have.

We can substitute the values we know into this formula to give an equation. The arc length is 115. We don’t know the value of 𝜃, so we’ll leave that as it is. And the radius of the circle is 50, which means that the diameter, which is twice the radius, will be 100. We, therefore, have the equation 115 equals 𝜃 over 360 multiplied by 100𝜋. And we can solve this equation to determine the value of 𝜃. First, though, we can simplify if we wish. On the right, the fraction 100 over 360 can be simplified to five over 18 by dividing both the numerator and denominator by 20.

To solve for 𝜃, we need to divide both sides of this equation by five 𝜋 over 18. And we recall that in order to divide by a fraction, we multiply by the reciprocal of that fraction. So, we have 115 multiplied by 18 over five 𝜋 is equal to 𝜃. Again, we can simplify a little if we wish by canceling a factor of five from the numerator and denominator to give 𝜃 equals 23 multiplied by 18 over 𝜋.

Evaluating this on a calculator gives 𝜃 equals 131.78029. And we then need to round this value to the nearest tenth, which gives 131.8 degrees. This gives the central angle of the sector. But remember, by definition, it also gives the measure of the arc. The length of the arc is 115 units. The measure of the arc is 131.8 degrees.

Let’s now look at another example in which we’ll combine our knowledge of arc length with some other mathematical skills.

Rectangle 𝐵𝐶𝑀𝐷 is drawn inside a quarter-circle, where 𝐵𝐷 equals nine centimeters and 𝐵𝐶 equals 12 centimeters. Find the length of arc 𝐴𝐵𝐸, giving the answer in terms of 𝜋.

So, we’ve been asked to find the length of the arc 𝐴𝐵𝐸. We know that we have quarter of a circle, which means that this arc length will be quarter of the circle’s full circumference. So, it’s equal to one-quarter multiplied by 𝜋𝑑. We can also think of that value of one-quarter as 90 over 360 if we wish because this quarter-circle is also a sector of a circle with a central angle of 90 degrees.

In order to answer this problem then, we need to know the diameter or perhaps the radius of this circle. Instead, we’ve been given some other measurements in the question. 𝐵𝐷 is nine centimeters and 𝐵𝐶 is 12 centimeters, but neither of these are the radius or diameter. We can identify three radii of this circle in the diagram: the lines 𝑀𝐴, 𝑀𝐵, and 𝑀𝐸. Now, as 𝐵𝐶𝑀𝐷 is a rectangle, we know that the length of the line segment 𝐶𝑀 will be the same as the length of the line segment 𝐵𝐷. So, it’s also nine centimeters.

If we look carefully at the figure, we now see that we have a right triangle, triangle 𝐵𝐶𝑀, in which we know the lengths of two of the sides; they’re 12 centimeters and nine centimeters. And the third side is the radius of the circle, which we wish to calculate. If we know two sides in a right triangle and we want to calculate the third, we can do this using the Pythagorean theorem, which tells us that the sum of the squares of the two shorter sides, 𝑎 and 𝑏, is equal to the square of the hypotenuse. So, this is often written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared.

In our triangle, the hypotenuse is the radius of the circle because it’s the side directly opposite the right angle. So, we have 𝑎 and 𝑏 as nine and 12 centimetres and 𝑐 as 𝑟 centimeters. We can, therefore, form an equation, 𝑟 squared equals 12 squared plus nine squared. And we can solve this equation to find the radius of the circle. 12 squared plus nine squared, that’s 144 plus 81, is 225. 𝑟 is, therefore, equal to the square root of 225, which is 15.

Now, you may actually have already realized this because nine, 12, 15 is an example of a Pythagorean triple. In fact, it’s an enlargement of the three, four, five Pythagorean triple with a scale factor of three. And the three, four, five triple is probably the most easily recognizable. In any case, we now know the radius of the circle; it’s 15 centimeters. We can, therefore, calculate the diameter of the circle by doubling this value. The diameter is 30 centimeters.

So, all that remains then is to substitute this diameter into our formula for the arc length, which remember was one-quarter multiplied by 𝜋𝑑. We have one-quarter multiplied by 30𝜋. And we can then simplify by canceling a factor of two from the numerator and denominator. We’re left with 15 over two 𝜋, which we can write using decimals, if we wish, as 7.5𝜋.

The question asks us to leave our answer in terms of 𝜋. So, we can include the units of centimeters and we have our answer to the problem. The length of arc 𝐴𝐵𝐸 is 7.5𝜋 centimeters. Notice that at no point in this question did we need a calculator. We left our answer in terms of 𝜋. And in our work with the Pythagorean theorem, we were working with a Pythagorean triple.

We’ve now seen three examples of calculating arc length. Let’s consider an example where we’re asked to calculate the area of a circular sector.

The area of a circle is 160 square centimeters and a central angle of a sector is 71 degrees. Find the area of the sector giving the answer to two decimal places.

In this problem, we have a circle whose area we know and a sector with a central angle of 71 degrees whose area we wish to calculate. We recall that in order to calculate the area of a sector with a central angle of 𝜃 degrees, we take the area of the full circle, 𝜋𝑟 squared, and multiply it by the fraction of the circle that our sector represents; that’s 𝜃 over 360.

In this problem though, we don’t need to calculate the area of the circle using 𝜋𝑟 squared because we’ve been given it. So, we can just substitute the relevant information. 𝜃 is 71 degrees. So, we have 71 over 360 multiplied by the area of the circle which is 160. We could evaluate this on our calculators or we could first simplify by canceling a factor of 40 from the numerator and denominator to give 71 over nine multiplied by four.

Now, we can evaluate and it gives 284 over nine or 31.5 recurring. Rounding to two decimal places and our value is 31.56. And the units for this area will be the same as the units for the area of the original circle; they’re square centimeters. So by multiplying the area of the circle by the fraction of the circle that this sector represents, we’ve found that the area of the sector to two decimal places is 31.56 square centimeters.

Let’s now consider one final example involving the areas of sectors.

In this figure, the diameter of the larger circle is 41 centimeters and both circles have the same center. Determine, to the nearest tenth, the area of the shaded part.

So in this question, we have two concentric circles. That means they have the same center. And we’re looking to find the area of the shaded part, which looks a little bit like part of a ring or donut shape. There are lots of different ways that we could approach this question. We need to think carefully about what the most efficient approach is. We have these two lines in pink, which are dividing each circle into a minor segment and a major segment.

I think the easiest way to approach this problem is as the difference between the area of the major sector of the large circle, which is outlined in pink, and the major sector of the smaller circle, which is outlined in green. We know that the area of a sector can be found by multiplying the area of the full circle, 𝜋𝑟 squared, by the fraction 𝜃 over 360, where 𝜃 is the central angle of the sector. Both of our sectors share a common central angle which we need to calculate.

We’ve been given the measure of the minor arc of the larger circle; it’s 134 degrees. And we recall that the measure of an arc is equal to the central angle of its sector. So, the central angle of the minor sectors is 134 degrees. The central angle of the major sectors then is 360 degrees minus this value which is 226 degrees.

Next, we need to determine the radius of each of our circles. We’re given that the diameter of the larger circle is 41 centimeters. So, its radius will be half of this; that’s 20.5 centimeters. Looking at the figure, we can work out the radius of the smaller circle as the difference between 20.5 centimeters and the measurement we’re given of 12.3 centimeters, which is the difference between the radii of the two circles. The radius of the smaller circle is, therefore, 8.2 centimeters. And now, we have all the information we need in order to calculate this area.

The area of the larger major sector is 226 over 360 multiplied by 𝜋 multiplied by 20.5 squared. And the area of the smaller major sector is 226 over 360 multiplied by 𝜋 multiplied by 8.2 squared. If we wish, we can factor this calculation by 226 over 360 multiplied by 𝜋 before evaluating. Using a calculator, we obtain 696.21410. And rounding to the nearest tenth, we have our answer to the problem. The area of the shaded part of the figure, which is the difference in the areas of the major sectors of the two circles, is 696.2 square centimeters.

Let’s now summarize some of the key points from this video. Firstly, an arc is a portion of the circumference of a circle, and a sector is a portion of the circle itself enclosed by two radii and an arc. The measure of an arc is defined to be the central angle of the sector, here represented using the Greek letter 𝜃. To calculate an arc length, we multiply the full circumference of the circle by the fraction of the circle that we have, giving 𝜃 over 360 multiplied by 𝜋𝑑.

We calculate the area of a sector in a similar way, this time multiplying the area of the circle 𝜋𝑟 squared by the fraction of the circle we have, 𝜃 over 360. Finally, we must remember that the formulae used in this video can only be applied if the central angle 𝜃 is measured in degrees.