If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Area of a sector

A worked example of finding the area of a circle's sector using the area of the circle and the central angle of the sector. Created by Sal Khan.

Want to join the conversation?

  • leafers ultimate style avatar for user Janet Bockey
    At where did he get nine times nine from? Also he divided the 350/360 by ten should he divide the other side by ten also?
    (16 votes)
    Default Khan Academy avatar avatar for user
    • hopper happy style avatar for user Alois
      Hey Janet, 9*9 is the same thing as 81.

      With fractions, if you divide the numerator by 10,
      you divide the denominator by 10 as well.

      You don't need to divide the other side,
      it's just simplifying a fraction and still the same number after all.
      (84 votes)
  • leaf green style avatar for user orshick
    I understand the lesson in the video, but the mastery question I got for this topic ( https://puu.sh/jJNcU/bfa5ed77f8.png ) uses 2pi as the number of radians in the whole circle, which is not explained in the video, and I have no idea where the "hints" answer got it from. Is this question assuming I know something that wasn't taught, or am I missing something obvious here?
    (35 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Kevin Finn
    How can you do this if you only have the arc measure and radius and it doesn't show the central angle measure?
    (10 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Just Keith
      If you know the arc length and the radius, then the angle that is subtended by the sector is θ = L / r
      where L= arc length and r = radius
      (Angle in radians, of course.)

      Thus, the area of the sector would be:
      A = (θ / 2π) (π r²)
      A = ½ θ r²
      Now, we plug in the formula for θ
      A = ½ (L/r) r²
      A = ½ r L
      (2 votes)
  • duskpin ultimate style avatar for user wonderwoman
    so the formula is "area of the sector divided by total area of the circle equals degrees of the central angle divided by total degrees in a circle" ?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Dominic Nguyen
      If you're asking for the area of the sector, it's the central angle of 360, times the area of the circle, for example, if the central angle is 60, and the two radiuses forming it are 20 inches, you would divide 60 by 360 to get 1/6. And solve for area normally (r^2*pi) so you would get 400*pi, than divide by 6, you would get around 209.
      Formula would be: (central angle/360)(r^2*pi)
      Side note, circumference of a sector would be the same formula except replace r^2*pi with formula for circumference
      Hope this helps
      (15 votes)
  • blobby purple style avatar for user Toodle
    At , can somebody explain the part he multiplied 350/360 * 81pi ? Why can't he multiply it like 350/360 * 1/81 ?
    At , why did he multiply 81pi for both side and not other number?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • starky tree style avatar for user Ina
      1. He could have multiplied by 81/1 π which is equal to (81/1) x π = (81x1) x π = 81 π . He could not have multiplied by (1/81)π as you suggest. (1/81) π = (0.01234567901234567901234567901235 x π) and is not the same thing.

      2. He did this because we want to calculate what "area of the sector" is. This means that we should have "area of the sector" on one side of the equation and the rest on the other side of the equation as we are looking for the answer to "area of a sector = ?"

      By multiplying both sides by 81π he cancels out he 81π on the left side of the equation below the "area of the sector" and moves it only to he right side of the equation so we are left with the clean "area of the sector =", which is what we are looking for on the right.

      You can also think of this step in this way:

      81π x area of a sector : 81π

      Every time you multiply something and then divide it by the same thing these steps cancel each other out, so you can just remove them from the equation just like 2 x 10 : 2 = 20 : 2 = 10 ... you see you don't even have to go through the trouble to multiply and divide if it is the same thing as the rest will just stay the same. ;)

      If this is hard for you I suggest you review the chapter on fraction. It has videos that explain how this works and also some practices for you:

      Also make sure you learn these properties found under this link, they are very important to know: https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-arithmetic-properties/v/order-doesn-t-matter-when-purely-multiplying
      (5 votes)
  • spunky sam green style avatar for user Not Friedrich Gauss
    So not only does that ratio give you the arc length but area of the angle subtended by the arc aswell?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • starky tree style avatar for user kai.mayahara
    How do you find the possible radii and central angles when you know the perimeter of the sector and the area of the sector?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user sanjanakabad13
    At , Where did he get the 9.9 from? And why 35? Wasn't it 350?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Anton Örjenfelt
    Hi. Please look at this image: http://sv.tinypic.com/r/29p3pyp/9

    Observe angle α. With angle α and vertex O as center I draw the circle sectors AB and CD. How do I prove that the circular sectors OAB and OCD are similar?
    I know that:

    If α = angle in degree's. And 2πr = Circumference

    b1 = (α/360°)(2πr1) , b2 = (α/360°)(2πr2) therefore
    b1/b2 = r1/r2 <==> b1/r1 = b2/r2 etc.

    Is this reasoning correct?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • cacteye blue style avatar for user Lindgren, Josh
    I am currently stuck.
    For example, I keep getting the answers for Area of a Sector (https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-sectors/e/areas_of_circles_and_sectors) wrong, and it tells me two different things. Most of the time I get the sector equation being Angle(sector)/360 = Area(sector)/Area(circle). However, sometimes I also get the sector equality of Angle (sector)/2π = Area(sector)/Area(circle). The difference is in the 2π and the 260. On a further note so you can figure out what I am getting at, I have seen this in the following problem:
    A sector with a central angle measure of 7π/4 (in radians) has a radius of 16cm. 
    What is the area of the sector?

    So what I did, like I had done in multiple other problems, was divide the angle of the sector, (7π/4) by 360. But when I found the answer to be incorrect, I looked through the help section of the question and noticed the 2π in the place of the 360.

    Can someone enlighten me as to how I am supposed to discern whether to use the 360 degree input or the 2π input underneath the sector angle portion of the Sector equality statement? Thanks.
    (3 votes)
    Default Khan Academy avatar avatar for user

Video transcript

A circle with area 81 pi has a sector with a 350-degree central angle. So this whole sector right over here that's shaded in, this pale orange-yellowish color, that has a 350-degree central angle. So you see the central angle, it's a very large angle. It's going all the way around like that. And they ask us, what is the area of the sector? So we just need to realize that the ratio between the area of the sector and the total area of the circle. And they tell us what the total area is. It's 81 pi. And 81 pi is going to be equal to the ratio of its central angle, which is 350 degrees, over the total number of degrees in a circle-- over 360. So the area of the sector over the total area is equal to the degrees in the central angle over the total degrees in a circle. And then we just can solve for area of a sector by multiplying both sides by 81 pi. 81 pi, 81 pi-- so these cancel out. 350 divided by 360 is 35/36. And so our area, our sector area, is equal to-- let's see, in the numerator, we have 35 times-- instead of 81, let's see, that's going to be 9 times 9 pi. And in the denominator, I have 36. Well, that's the same thing as 9 times 4. And so we can divide the numerator and the denominator both by 9, and so we are left with 35 times 9. And neither of these are divisible by 4, so that's about as simplified as we can get it. So let's think about what 35 times 9 is. 35 times 9, it's going to be 350 minus 35, which would be 315, I guess. Did I do that right? Yeah, it's going to be 270 plus 45, which is 315 pi over 4. 315 pi over 4 is the area of the sector.