If a circle has radius 5, then the circumference is approximately...

Compound Perimeter

Key Stage 3

Maths

Circles

If the diameter of a circle is 20. What is the circumference?

If the circumference of a circle is 14 pi. What is the radius?

﻿Hello, I'm Mr. Coward. And welcome to today's lesson on compound perimeter. For today's lesson, we'll need a pen and paper or something to write on and with, and a calculator. If you can please take a moment to clear away any distractions, including turning off any notifications, that would be great. And if you can, try find a quiet space to work where you won't be disturbed. When you're ready, let's begin. Here's the time for try this task. Use the sizes shown on the right. Find the perimeter of these shapes. Here, we have lots of the diameters, lots of arcs and these extra little bits at the bottom. You need to try and work out the perimeter of these shapes. Maybe if you finished that that okay. Then maybe try and make a shape of you own and find the perimeter of that. Pause the video and have a go. Pause in three, two, one. Welcome back. Now, let's talk about a few of the things that I worked out. Let's first work out all of these arc lengths. So this one, well, that's got a diameter of 20. The full circle would have us a conference of 20 pi. 1/2 the circle, would be 1/2 of that. 1/2 of 20 pi, 10 pi. This one, 16. The full circle would have a circumference of 16 pi. So 1/2 the circle, this one. The full circle would have a circumference of 12 pi. So 1/2 the circle, and finally, the last one, four pi. If we know that that's 10 pi and that one is six pi, We've got that and that. But we don't want to have these lengths here. Let's compare the difference there. That one's 12, going across and that one 14 going across. Sorry. 20 going across and 12 going across. So that is a difference of eight. The difference between the diameters is eight. That length there is half of the difference and that's half of the difference. So 1/2 of eight is four, which means that that length there is four. Total perimeter is 10 pi plus 6 pi plus four plus four, which we can just leave like that. When we're adding on a value with pi and a normal number, we do not add together the number in front of the pi. The 16 and the eight, we leave them separate. 16 times 2 plus 1 is not the same as, we'll do plus a bigger number to make it more obvious. 16 times 2 plus 13 is not the same as 16 plus 13, 29 times 2. They are not equal to each other. That 16 times pi, so we do that first and then we add eight. We can't mix this number and this number. So anyway, that is the perimeter of this shape. What about this shape? Well, we have 10 pi, four pi, four pi. And what are all those lengths added together? What is the difference between the big diameter, 20, and two of these, 8 plus 8, 16. So what we've got is we've got 20 plus 16, which is 4. It doesn't matter how long that base or how long that base or how long that base. The bit that does matter is those three bits add up to four. Our total perimeter is 10 pi plus 4 pi plus 4 pi, 18 pi plus 4 centimetres. And final one, we have eight pi, four pi. And how long is that length there going to be? That is eight. And that is 12. Sorry, that is 16. 16 minus 8 is 8. So that length is eight. So we have 12 pi plus 8 centimetres. There's a lot going on there and it was quite tricky, this task. And you might have worked out your answers in decimals, which is fine as well. So it might be worth you checking on your calculator if this gives you the same answer as your decimals. What we're going to do now is we're going to find the perimeter of these shapes. I have to use one shape for now and then I'm going to give your turn. Now what is the perimeter? Do you remember what the perimeter means? That means, going the length of the outside. The length of the boundary of the shape. Going there and there and there. So we've got three lengths to consider for the perimeter. What is this length going to be? This length here. That is going to be the same as that, it's going to be six. And we don't know what the units are. So I'm just going to leave it as six. Why is that six? Because the radius is the same at any point on the circle. The radius is always the same. No matter where you are, the radius is the same. We just need to find this length. Well, how do we find that length? That length, is just an arc length. And we've learned in previous lessons how to find an arc length. How do we find the arc length? We find the circumference of the full circle. If it was a full circle, what would the diameter be? With the radius as six, the diameter would be 12. So we do 12 times pi, which gives us 12 pi. And that is the circumference, oh, sorry, of full, I'll just draw a circle, full circle. That would be the circumference of the full circle. But what fraction of the circle do we have? We have 102 out of 360 degrees. So we times that by 12 pi and we would get our arc length. That I would have to work out on a calculator, but fortunately I prepared early. So that gives me 10.68, to two decimal places. To work out my perimeter, I would add those two lengths to that length. That plus that, plus that. So 10.68, which gives me 22.68. So, your turn. Pause the video and have a go. Pause in three, two, one. Okay, welcome back. Hopefully, you got the circumference of the full circle is 16 pi. And then you got what fraction of the circle it was. And times that by 16 pi to get that arc length, which is a 30.72. And then you added on your two radii. So 8 plus 8, 16. 16 plus 30.72 gives you 46.72. So really well done if you've got that correct. Find the perimeter of these shapes. And this time we're going to give our answers in exact form. What fraction of the circle is this? It's 90/360 or 1/4. We need to work out this length. The full circle would have circumference of 24, 12 times 2, to get the diameter. Diameter times pi. Let me write where I got that from. 12 times 2 times pi. 12 times 2, to get the diameter, and times by pi to get the circumference. And now we would find the perimeter of that bit, that arc length. So we find the arc length, which is 1/4 of this number. What is 1/4 of 24 pi? 6 pi. So now, we need to add on our 12 and our 12. So we do 24 plus 6 pi. And that's from 12 plus 12. And that would be my final answer. Let's go over those steps. Find out the circumference of the full circle. Find out what the arc length is and then do two radii plus arc length. It's your turn. Pause the video and have a go. Pause in three, two, one. Okay, welcome back. Hopefully, you got the full circumference would be 16 pi. The arc length, 3/4 of a circle, would be 3/4 of 16 pi, which is 12 pi. Now we've got that's 12 pi. We have to add on our two radii. So 8 plus 8, which is 16. And you got the answer of 16 plus 12 pi. And it doesn't matter if you add 12 pi plus 16, they are the same thing. And over here, 6 pi plus 24 is the same as 24 plus 6 pi. Slightly more complicated shape now. Well, can you see this? If I do this line here. I've got 1/4 of a circle. What shape do I have here? That length is 12 centimetres. So that's 12 centimetres. there radius be? 12 centimetres. Which means that this length is 12 centimetres. Which means I've got a square and 1/4 circle. Find the perimeter of the shape. I need to find this length here. My full circle would have a circumference of 24 pi because the radius is 12, so the diameter is 24. 24 times pi, 24 pi. Diameter is 24 times pi. Part two. What would this length be here? What's 1/4 of the circle? So it'd be 1/4 of 24, which is 6 pi. That's 1/4 of 24 pi. Now finally, we've got our 6 pi plus our other lengths. So we've got 6 pi there and what's that length there going to be? 12, 'cause it's a square. So 6 pi plus 12 plus 12 pus 12, plus another 12. 6 pi plus 12 times 4. So 6 times pi plus 12 times 4. So you get 6 pi plus 48. And that's in centimetres in there. Your turn. And I'm just going to give you a little hint. I'm just going to draw this little line in here. You have a semicircle and a rectangle. Find the perimeter. Pause in three, two, one. Okay, welcome back. Hopefully, you got that this is a semicircle with diameter, 30. The circumference of the full circle would be 30 pi. Because the diameter is 30, not the radius. So it's just 30 times pi. That length is 30 pi. No, sorry, that length is not 30 pi. The full circle would be 30 pi and that's 1/2 of that 30 pi. So 1/2 of 30 pi is 15 pi. So we've got 15 pi. That length is 12, that is 30 and that length is 12. So that gives us 15 pi plus, that's 24, 24 plus 30, 54. So really well done if you've got that correct. That is not an easy question by any means. Oh, last one. I thought it was the time for the independent task but it's not This one is really tricky. The first thing I'm going to do, is I'm going to imagine we had a full circle. Like this. That full circle has a diameter of 32. What would that circumference be? 32 pi. Where did I get that 32 from? 16 plus 16. We've got that length, is 32. What would 1/2 of that be? So 1/2 a 32 pi is 16 pi. We've got 16 pi there. Then we've got this length. What would that length be? The full, little circle would be that. So the full circumference would be 16 pi. So 1/2 of that is eight pi. So that there is eight pi. Now, can you see how that length is the same as that length? Because they would be, if you drew two circles in, circle there, and a circle there, you'd see how that circle and that circle are the same. So that is 16. That's a circle, we're down to 16. So that means that's eight pi. Our full perimeter is, 16 pi plus 8 pi plus 8 pi, which is 32 pi. We can just add our pis together. You get 10 pi plus 5 pi it'd be 15 pi. If you add 12 pi plus 9 pi, that'd be 21 pi. If you add 2 pi plus 3 pi, that'd be 5 pi. Just as if you had this, say you had 5 times 3 plus 4 times 3. That is equal to 9 times 3. 5 plus 4 is 9. So just using the distributive property, that's why we can collect our pis together. So now it's time for the independent task and know the answers are given in exact form, into two decimal places. Pause the video to complete your task and resume once you've finished. Okay, welcome back. Here are my answers. Now it's just time for the explore task. We've got a second hand with length 12 centimetres, a minute hand with length 10 and an hour hand with length six. How far does each hand travel in one minute, one hour, 12 hours? After how much time does each hand travel six pi centimetres? That's something to think about. I do have a hint for ya but have a go first. Pause the video to complete your task and resume once you're finished. If you need a hint, that length is 12. That's like a radius of 12. So what would the diameter to be? This length would give us, this is going to be an awful, I am very sorry for this. Look how bad that is. That would be a circle with radius 10. This one, this would give us a little circle with radius six centimetres. So now, what fraction of the circle would be travelled by each hand in a minute? The second hand would go all the way around. The minute hand would move from there to there. 1/60 of the circle. How many degrees is that? The hour hand, what fraction of the circle is that? Well, in an hour, it moves that far in an hour. It moves 1/12 of the circle in an hour. So in a minute, it moves 1/60 of 1/12. Which is what? I've given you some hints. Hopefully, that will get you started. And then come back once you've finished. So here are our answers. So what do we have? Let's start with the second hand. The second hand going all the way around would be 360. So just be the circle of radius 12. So that means the diameter is 24. So it'd be 24 pi going all the way around, 24 pi centimetres. What about the minute hand? That would go 1/60 of the circle. 1/60 of the circle, what would the full circle be? Would be 20 pi. It's 1/60 of 20 pi, which would be a 1/3 pi. And what about the hour hand? That goes 1/60 of 1/12. Because in an hour, it goes 1/12. So in a minute, it goes 1/60 of that amount. So it's just 1/720. It goes 1/720 of a circle with conference 12 pi. 1/720 of 12 pi would be our answer. So we'd get these. Okay, then an hour. You could have worked out all again or what I did is I timesed all these distances by 60. 'Cause it would travel 60 times as far in an hour. And then in 12 hours, it would travel 12 times further than that. Now we have all the distances for a certain amount of time. After how much time does each hand travel six pi centimetres? In a minute, the second hand, just so you know, this is the second, that's the minute and that's the hour. The second hand travels 24 pi. Six pi is 1/4 of 24 pi. So 1/4 of 60 seconds would be 15 seconds. I got that. Then this one, 18 minutes. If it travels 1/3 pi, how many 1/3 pis are in six pi? 18. So take 18 minutes. For the hour, it travels pi in an hour. So travel six pi in six hours. Really well done if you got that. It was quite a tricky task, this. I'm very impressed if you managed to work some of this out. Not all of it, but just some of it. That is all for this lesson. Thank you very much for all your hard work. And I look forward to seeing you next time, thank you.

Compound Perimeter

Compound Perimeter

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UnitMaths / Circles

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