May 12, 2026

Added: 2026-05-19

[00:00:00]So, we do areas and lengths of sectors.

[00:00:03]And what we mean by a sector is a portion of a circle that has been cut off where we know the angle and we know the radians perhaps.

[00:00:15]And we're going to have to be asked to cut find a couple of things. So, we know that we're going to ask to The two things we might get asked are going to be the area of the sector, the arc length of the sector, and the perimeter of the sector.

[00:00:36]All right. So, the area obviously of the sector equal depending on which is the big or the the small sector of course because you might be asked to find the small bit, um which is here. Obviously, the small bit is inside here. I'm just going to try and do it a color here. That would be a minor arc sector. And of course, the outside bit would be the major arc. So, you could be asked to find either area depending on which what the question says.

[00:01:12]Mainly, it's the smaller bit we're interested in.

[00:01:16]But of course, how would you find the other angle? Well, we know these two all two angles here would add up to 360. So, this would either be 2π minus uh 2π minus uh theta or if it was degrees, it'd be 360 minus theta.

[00:01:40]So, the one problem is here, you won't get given the perimeter formula. You only get given the area of a sector and the arc length. And of course, the perimeter is going to be Well, what is it? It's the arc length, this bit here which we can work out with the formula we're given plus r plus R So the question might be work out the area of a sector. They'll give you a sector and they'll give you an angle.

[00:02:12]Let's say 32 degrees and this is 10. You got to find the area. Well, we know the area is degrees, so it's going to be 32 over 360 * pi r squared.

[00:02:29]Whatever that equals and that equals 26.18 units squared.

[00:02:41]And it might be an arc length, of course, in which case we'd say theta over 180 * pi * R and that would be cool.

[00:02:56]5.8 Sorry, 5.59 units. They usually give you centimeters or whatever it is, inches.

[00:03:05]Let's put units down.

[00:03:07]That's if it was degrees. Of course, they might give you radians, might they?

[00:03:11]So, we've got radians the same thing.

[00:03:13]Let's say that this is pi over three and they give you this is seven.

[00:03:20]So, we know that area equals uh formula is 1/2 >> [snorts] >> theta pi over three * r squared * 7 squared and that equals uh 25.66 units squared.

[00:03:46]And the arc length of this one would be um theta pi over three times r times the radius which equals 7.33 units.

[00:04:07]Of course, if they asked you to find the radius, then we'd add obviously 10 + 10 to the arc length for the first one and we'd add 7 + 7 for the to the arc length for the second one.

[00:04:23]Of course, the other type of problem they might give you, they might give you the area and you have to find out the angle. So, we'll look at that one. Let's say this is five and we know that area equals 50 units squared.

[00:04:43]So, we know that theta Well, let's have a look. Let's say theta is in um, radians. We know it's a half theta r squared. So, we know that 50 equals 1/2 times theta times 5 squared.

[00:05:01]And we mess around with it, multiply both sides by two, we get 100 equals 25 theta and theta equals four.

[00:05:12]And that'll be four radians.

[00:05:17]If we did it in degrees, we'd get the same similar answer cuz four radians is going to have a a degree answer. So, we know that 50 equals um, theta which is going to be theta over 360 times pi times radius squared, 25 squared.

[00:05:37]And Sorry, 25 not times 25 squared.

[00:05:42]Um, okay. So, therefore we divide by 25 and we get two = theta * pi over 360.

[00:05:51]360 * 2 is 720 = theta * pi and then we divide by pi.

[00:05:59]720 over pi.

[00:06:01]Now those got pi in the answer, the answer is still radians and the answer is 229.18° 229.18° and just to prove they're the same we do have that formula for converting degrees to radians, don't we? So 4 radians, if we multiply by 180 and divide by pi that will give us the answer in degrees.

[00:06:29]This is radians to degrees.

[00:06:35]And of course it's the other way around for degrees to radians. So 4 * 180 is actually 720 over pi and look we got the same calculation so we know must equal 229.18° exactly the same thing.

[00:06:52]A cryptic kind of question would be to give the perimeter of a sector.

[00:07:01]Let's say this is R R we know the angle is say 40° and we know the area sorry the perimeter what should we say? 21 cm so we know that the perimeter the 21 will be the arc length so it's going to be 40 over 180 >> [sighs] >> degrees * pi * the radius + 2 * the radius.

[00:07:39]So that's what we've got there. So, we know that 21 equals Now, we've got to work these both out, haven't we? So, 40 over 180 * pi that is 0.698. [snorts] r + 2r So, therefore, 21 = 2.698r and if we divide by 21 by 2.698 we should get the value of r equals 7.78 whatever the units are, 7.1 units.

[00:08:28]Cuz they could give you a question where they don't give you any information except the area and the arc length. We know that r is r, radius is r, and that's theta.

[00:08:44]And we know say the area is 32 cm squared and the arc length is say 10 cm.

[00:09:01]So, this means we're going to make two formulas. We know that area and we're going to work in radians here, so it's 1/2 theta r squared.

[00:09:12]So, there's one formula. That's the area formula.

[00:09:18]And then the arc length formula is going to be that 10 equals um theta * r.

[00:09:33]So, we've got two formulas here, and we have to um substitute for each one. So, what we could do, we could find what theta equals here. Well, theta equals 10 over R and we can actually go ahead and substitute it in right there.

[00:09:53]So, where we've got theta we're going to just replace it with 10 over R. So, we get 32 equals 1/2 10 divided by R times R squared.

[00:10:13]Now, of course, one of those R's disappears. So, we get 32 equals 10 over 2 times R. Well, that's quite easy, actually. 32 equals 5R.

[00:10:25]So, R equals 6.4.

[00:10:29]And therefore, theta will be 10 divided by 6.4 which equals 1.5625.

[00:10:40]So, let's say 1.56 rounded to two uh three significant digits.

[00:10:48]So, the next type of question they might ask you is to find the area of a shaded region in this sector. We've done these types of question before. We might say that that is five. The radius is five. The angle say is 45 degrees.

[00:11:09]Now, we've got to find the shaded region here.

[00:11:16]Well, of course, what we've got is an shaded region equals area of triangle Sorry, area of sector.

[00:11:32]Let's just do that a bit better. Let's do that underneath each other. Equals area of sector minus the area of the triangle.

[00:11:50]So, what's the area of the sector? We know the sector area of the sector is going to be theta over 360 cuz we're in degrees here * pi * the radius squared. Well, that's 5 squared.

[00:12:06]And what's the area of the triangle?

[00:12:08]We're going to use the 1/2 AB sin C. I always use 1/2 5 adjacent adjacent * sin 45°.

[00:12:19]All right. And we know that theta is 45.

[00:12:21]I didn't put that in.

[00:12:23]So, it's going to be 45 over 360.

[00:12:29]* 25 pi cuz 5 squared is 25 - 25 over 2 * sin 45. And we get our calculators out, work that out.

[00:12:44]And we get Let's do the first bit. Many decimals I can here. The area of the sector is 9.817.

[00:12:52]I think it's 9.817 47.

[00:12:57]It's 477. So, I'm going to put Oh, sorry about that. At least decimal points.

[00:13:01]Minus 9.739.

[00:13:08]978 And then we subtract those two.

[00:13:12]I get 0.077 78 units squared.

[00:13:29]On the final type of odds question you might get asked is a question where you've got donut shape like this and there's the center and you have got to find the area of this shaded region here.

[00:13:54]So, what you've got usually is that the line will extend to the middle and you're going to be given two radiuses, say two and just suppose this length here is three.

[00:14:08]So, you've got a little circle radius equals two, little circle.

[00:14:17]And then a big circle radius five.

[00:14:22]But, what you've got is that the angle in between the two is going to be the same. So, let's say that theta equals pi over three again.

[00:14:35]So, to find the area of the shaded bit, you do the area of the big sector.

[00:14:40]So, area of big would be um 1/2 theta pi over three times the big radius, which is five squared.

[00:14:59]And then the area of the small would be the same formula, 1/2 pi over three but this time times the smaller radius two squared. And we just subtract the answers.

[00:15:15]So, the top answer would be 13.089 0899, so I'm trying to hang on to as many decimal places as I can.

[00:15:27]Um, and I'm going to get it wrong. I'm going to put dot dot dot which means I'm I'm hanging on to the other decimal places.

[00:15:34]And then the other answer is going to be 2.0943.

[00:15:45]And then if I subtract the two shaded area Here's the big one minus the small so 13.089.

[00:15:56]Minus 2.0943.

[00:16:00]Dot dot dot.

[00:16:01]Equals 10.99 unit squared.

[00:16:13]And that's it. That's a basic summary of everything and um that should be fine. Good luck.