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### Course: Geometry (all content)>Unit 14

Lesson 6: Sectors

# 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?

• 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?
• 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.
• 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?
• Some of the geometry section overlaps with the trigonometry section, which may spark some confusion. As a rule of thumb:
You can use this identity to convert between radians and degrees.
• How can you do this if you only have the arc measure and radius and it doesn't show the central angle measure?
• 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

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
• 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" ?
• 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
• some of the questions in the lesson have nothing to do with this video
• This is to make sure you completely understand the subject. If all the questions were exactly the same type as the ones in the video, you wouldn't be getting the full picture. All the information is there: You just have to manipulate it to get your answer.
(1 vote)
• 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.
• Using the 360 degree input is when we are solving in degree form or in terms of degrees
the 2π input is when we are solving in radian form
Notice how the question mentions the central angle measure in terms of radians.
If it put the central angle measure in degrees (Which would be 315 degrees.. 7π/4 = 315°) , that's when you use the 360 degree input.

This was answered 3 years late and you probably have figured it out by now but I hope that this was helpful in any way! :)
(1 vote)
• 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?
• 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
• So not only does that ratio give you the arc length but area of the angle subtended by the arc aswell?
• There are two different ratios, but they are related, and the correct term would be sector area formed by the arc. You cannot solve them both at the same time, but with two different ratios. This may be what you meant.