Lesson Video: Areas of Circular Sectors Mathematics

In this video, we will learn how to find the area of a circular sector and solve problems that relate this area to the arc length and perimeter of the sector.

17:13

Video Transcript

In this video, we will learn how to find the area of a circular sector and solve problems that relate this area to the arc length and perimeter of the sector. We recall first that an arc is a portion of the circumference of a circle. A sector is a portion of the circle itself formed by two radii. That’s the lines that connect the center of the circle to its circumference and an arc. It’s sort of like the surface of a wedge of cheese cut from a wheel or a slice of pizza. The angle at the center of this wedge, the angle between the two radii, which we often denote using the Greek letter 𝜃, is called the central angle of the sector.

If this central angle 𝜃 is less than 180 degrees, we call the sector a minor sector. That’s because there will also be another sector in the circle whose central angle will be greater than 180 degrees. This sector’s bigger, so we call it the major sector. Now finding a formula for the area of a sector is fairly straightforward, but it does depend on whether we’re working in degrees or radians for the central angle.

Let’s consider a formula in degrees first of all. We know that the area of a full circle is calculated using the formula 𝜋𝑟 squared. But when we’re working with a sector, we only have a portion of the full circle, and so we only want a fraction of its area. The fraction of the circle we have is determined by the central angle. It will be 𝜃 over 360 because there are 360 degrees in a full turn. So to find the area of a sector, we simply multiply the area of the full circle by the fraction that the sector represents. So we have 𝜃 over 360 multiplied by 𝜋𝑟 squared.

So that’s great if we’re working in degrees. But if we’re working in radians, we need something different. In our first problem, we’ll consider how we can derive an alternative formula to use if instead the central angle is measured in radians.

Write an expression for the area of a sector whose arc’s measure is 𝜃 radians, knowing that the expression for the area of a sector measuring 𝜃 degrees is 𝜋𝑟 squared 𝜃 over 360.

So we’re reminded of the formula we can use to calculate the area of a sector when the central angle is given in degrees. And we’re asked to use this to determine a different formula we can use when the angle is given in radians. We should recall that when we’re working in radians, a full turn, which in degrees is equivalent to 360 degrees, is two 𝜋 radians. So we can take the formula that we know for the area of a sector in degrees, and we can replace the 360 in the denominator, which represents the 360 degrees in a full turn with two 𝜋. Doing so gives 𝜋𝑟 squared 𝜃 over two 𝜋. Now, of course, we can cancel a factor of 𝜋 from the numerator and denominator of this fraction, which leaves us with 𝑟 squared 𝜃 over two or, equivalently, one-half 𝑟 squared 𝜃.

So we’ve used the area of a sector in degrees to find an expression for the area of sector when the central angle is given in radians; it’s one-half 𝑟 square 𝜃.

So we now have the two formulae for determining the area of a sector for central angle measured in either degrees or radians. When answering a problem, we must make sure we choose the appropriate formula for the measure that has been used for the central angle. It will also be helpful at this point to recall the formulae for calculating an arc length both in degrees and radians.

In degrees first of all, the arc length is 𝜃 over 360 multiplied by 𝜋𝑑 or two 𝜋𝑟. It’s the fraction of the circle we have multiplied by the circle’s circumference. And in radians, the arc length is simply 𝑟𝜃. That’s because if we take the formula in degrees and then replace 360 with two 𝜋 and 𝑑, the diameter, with two 𝑟, we have 𝜃 over two 𝜋 multiplied by 𝜋 multiplied by two 𝑟. And of course, the 𝜋’s will cancel and so will the twos, leaving us with simply 𝑟𝜃.

We’ve got all the formulae we need. So now let’s consider some examples. The problems that we consider in this video will be applications of these results to questions which require an element of problem solving.

The radius of a circle is five centimeters and the area of a sector is 15 centimeters squared. Find the central angle, in radians, rounded to one decimal place.

In this problem, we have a sector of a circle. We know that the radius of the circle is five centimeters and we know that the area of the sector is 15 square centimetres. But we don’t know the central angle. We were asked to find the central angle in radians. So we need to recall the key formulae we need. When working in radians, the area of a sector is given by a half 𝑟 squared 𝜃, where 𝑟 represents the radius of the circle and 𝜃 represents the central angle. We can therefore use the information we’re given to form an equation. The area of the sector is 15 square centimeters, and the radius of the circle is five centimeters. So we have the equation 15 equals a half multiplied by five squared multiplied by 𝜃. We can now solve this equation to determine the value of 𝜃.

Firstly, we evaluate five squared, which is 25. We can then multiply both sides of the equation by two and then divide both sides by 25, giving 𝜃 equals two multiplied by 15 over 25. Two multiplied by 15 is 30, and then we can cancel a factor of five from both the numerator and denominator. So this simplifies to six over five. Six over five is equivalent to one and one-fifth or 1.2. And so we have our answer to the problem. The central angle of the sector in radians is 1.2 radians.

Notice that in this question it was important to use the area formula in radians. It would have been possible to instead use the area formula in degrees and then convert the answer from degrees to radians at the end. But that would require an extra step. And of course, we may forget to convert our answer.

Let’s now consider an example in which we use our knowledge of the area of a sector to find the area of a related region.

Find the area of the shaded part of the quadrant in the diagram in terms of 𝜋.

We haven’t been told whether to work in degrees or radians, so let’s choose to work in degrees. We know that the area of a sector whose central angle is 𝜃 and whose radius is 𝑟 is 𝜃 over 360 multiplied by 𝜋𝑟 squared. In this problem, the central angle is 90 degrees, so we have 90 over 360 multiplied by 𝜋𝑟 squared, which simplifies to one-quarter multiplied by 𝜋𝑟 squared or 𝜋𝑟 squared over four. This, of course, makes sense because we know that a quadrant is quarter of a circle, so the area of a quadrant will be quarter of the full circle’s area. For the area of the larger quadrant first of all then, we have one-quarter multiplied by 𝜋 multiplied by 25 squared. And for the smaller quadrant, it’s one-quarter multiplied by 𝜋 multiplied by 17 squared.

If we want, we can factor by 𝜋 over four giving 𝜋 over four multiplied by 25 squared minus 17 squared. 25 squared is 625 and 17 squared is 289. 625 minus 289 is 336. So we have 336𝜋 over four. And finally, we can cancel a factor of four from the numerator and denominator to give 84𝜋. The units for the radius were centimeters, and so the units for the area will be square centimeters. So we found that the area of the shaded part of the quadrant in terms of 𝜋 is 84𝜋 square centimeters.

Let’s now consider the problem where we’ll see how we can find the arc length of a sector if we know its area and we know its central angle.

The area of a circular sector is 1888 square centimetres and its central angle is 1.7 radians. Find the arc length of the sector giving the answer to the nearest centimeter.

In this problem then, we have a circular sector. We know its area is 1888 square centimeters and we know its central angle, which we can call 𝜃, is 1.7 radians. We want to calculate the arc length of the sector, which we often denote using the letter 𝑠. The key thing to note is that the central angle of the sector is measured in radians. So the formulae that we’re going to use for both the area of the sector and the arc length must be the formulae in radians. Those are a half 𝑟 squared 𝜃 for the area of the sector and 𝑟𝜃 for the arc length.

In order to calculate the arc length then, we need to know both the radius and the central angle. Let’s use the information we were given about the area to form an equation. The area is 1888 square centimeters. We don’t know the radius, so we’ll keep the letter 𝑟. And central angle 𝜃 in radians is 1.7. We can now solve this equation to work out the radius of the sector, which we’ll then be able to substitute into our formula for the arc length. We can multiply both sides of the equation by two and then divide by 1.7 giving 𝑟 squared is equal to 1888 multiplied by two over 1.7, which as a decimal is 2221.176.

To solve for 𝑟, we take the square root of each side of this equation, taking only the positive value as 𝑟 represents a length. That gives 47.129, and we’ll keep this exact value on our calculator display for now. So we’ve now calculated the radius of the sector, and all that remains is to calculate the arc length. Using the formula arc length equals 𝑟𝜃, we multiply the value we’ve just calculated by the central angle of 1.7, which gives 80.119. We were asked to give an answer to the nearest centimeter, so we round down. Using the known area and the known central angle of this circular sector then, we’ve calculated that the arc length to the nearest centimeter is 80 centimeters.

In our final example, we’ll see how we can solve a different geometric problem by applying these techniques.

Three congruent circles with a radius of 43 centimeters are placed touching each other. Find the area of the part between the circles giving the answer to the nearest square centimeter.

So we want to find the area of the part between these circles. That’s this part here. We know that the circles are congruent, that’s identical, each with the radius of 43 centimeters. If we join the centers of the three circles, we have a triangle. Each side of this triangle consists of two radii. And so, in fact, this is an equilateral triangle with a side length of two times 43; that’s 86 centimeters. The area we’re looking to find is inside this triangle. The other areas inside this triangle, that’s these three areas here, are three congruent circular sectors. So our approach to finding the blue area is going to be to find the area of the triangle and then subtract the area of these three sectors.

The triangle, first of all. This is a nonright triangle, so we need to use the formula a half 𝑎𝑏 sin 𝑐, where 𝑎 and 𝑏 represent two sides of the triangle and 𝑐 represents the angle between them. Each of the side lengths of our triangle are 86 centimeters. And as it’s an equilateral triangle, all of the angles are 60 degrees. So we have a half multiplied by 86 multiplied by 86 multiplied by sin of 60 degrees. sin of 60 degrees is root three over two, so the area of the triangle simplifies to 1849 root three, which we’ll keep in its exact form for now.

Now, for the area of the sectors, the area of each sector working in degrees is 𝜃 over 360 multiplied by 𝜋𝑟 squared, where 𝜃 is the central angle. But we have three of these identical sectors, so we’ll multiply by three. That would be equivalent then to the formula 𝜃 over 120 multiplied by 𝜋𝑟 squared, as three over 360 is one over 120. So substituting the angle of 60 degrees and the radius of the sector which, remember, is 43 centimeters, we have 60 over 120 multiplied by 𝜋 multiplied by 43 squared. 60 over 120 simplifies to a half and 43 squared is 1849. So we have 1849𝜋 over two for the area of the sectors.

We can now go ahead and evaluate this using our calculators. Doing so gives 298.159. We’re required to give an answer to the nearest square centimeter, so we’ll be rounding down. We found then that the area of the part between the circles, which is the difference between the area of a nonright triangle and three circular sectors, is 298 square centimeters.

Let’s now review some of the key points that we’ve covered in this video. A circular sector is a portion of a circle formed by two radii and an arc. We use the Greek letter 𝜃 to denote the angle between the two radii, which is called the central angle of the sector. If the central angle is less than 180 degrees or 𝜋 radians, we have a minor sector, whereas if it’s greater than 180 degrees, that’s greater than 𝜋 radians, we have a major sector. The formula we use for calculating the area of a sector depends on whether the central angle is measured in degrees or radians. In degrees, it’s 𝜃 over 360 multiplied by 𝜋𝑟 squared, whereas in radians it’s a half 𝑟 squared 𝜃.

We also need to recall the formulae for calculating an arc length. In degrees, it’s 𝜃 over 360 multiplied by two 𝜋𝑟 or 𝜋𝑑. And in radians, it’s simply 𝑟𝜃. We often use the letter 𝑠 to denote the arc length. If we wanted to calculate the perimeter of a circular sector as opposed to just the arc length, we’d need to add on twice the radius. So we have that the perimeter of the sector is equal to 𝑠 plus two 𝑟. We can use these formulae to solve problems relating arc length and the area of a sector as well as problems in other geometric and real-world contexts.