What calculation would work out the shaded area?

What calculation would work out the total area?

A square of perimeter 80cm is attached to a semi-circle, they fit together with no overlap. What is the total area?

Compound area problems

Key Stage 3

Maths

Circles

If the circumference of a circle is 12pi, what is the area?

What is the area of this sector to 1 decimal place?

﻿Hello, I'm Mr. Coward. And welcome to the final lesson of the unit on circles. For today's lesson, we'll be looking at compound area problems. All you'll need for today's lesson is a pen and paper or something to write on and and with and a calculator. If you can, please clear away any distractions, including turning off any notifications. If you can, try and find a quiet space to work where you won't be disturbed. Okay. When you're ready, let's begin. Okay. So time for the, Try This task. Explain why the green area in each picture is the same. Can you calculate the green area? So pause the video and have a go. Pause in three, two, one. Okay. Welcome back. Now, hopefully, you have found that this is a circle with diameter equal to 10. Okay. That's 10. So the diameter that should be passing through the... I'm sorry. That diameter is equal to 10. So what is the radius? The radius is half the diameter. So the radius is five. Now. Why did they work out the radius? Well, I worked at the radius because I need the radius for my area. So to find my area Pi rxr, rxr 25. So we have 25 Pi. Okay. So that's the area of my circle, but that is not the shaded area. The shaded area is the difference between the square and the circle. The square, subtract the circle. There area of the square, 10x10. Subtract the area of the circle. And that gives me my green area. So that is the green area in exact form. So really well done. If you managed to work that out. Now, can you see how they're the same? Well, they're the same, because if you chop that into quarters and you imagine rotating that quarter, so that bit there goes there. Then, if you did it to all of them, you'd get that kind of shape in the middle and you'd have your quarters of your circles around the outside. Okay. So we're going to look at compound shapes now. And compound shapes are just more than one shape. Okay. Multiple shapes put together. In this case, more so, we have two shapes that have been taken away and put together. So we have a circle with a smaller circle inside that's been cut out. So to work out the area of this shaded part, we need to do find the big circle, the area of that, and subtract the area of the small circle. And that will give us this green shaded area. So area of the big circle. What is the radius? The radius is four of the big circle. So 4x4 Pi. Then what is the radius of the smaller circle? but it's not four, is it? That's the diameter. It's two. So the area of the smaller circle is 4 Pi. So what is the area of the shaded section? Well, I subtract my smaller area from my bigger area and that gives me an area of 12 Pi. I'm just going to leave it in exact form. Okay. So your turn. Pause the video and have a go. Pause in three, two, one. Okay. Welcome back. Now, hopefully you worked out the area of the big circle. It's got a radius of three. So it's 3x3 Pi, which gives you 9 Pi. And then, hopefully you worked out the area of one of the smaller circles. Now that has got a radius of 1.5. So you could do 1.5x1.5, which gives us 2.25 Pi as a decimal. Now we've actually got two of them. Can you see how we've got two of them there? So the area of both of those two is 4.5 Pi. So what we have to do is we have to do the full circle, 9 Pi take away 4.5 Pi. And that gives us our area. Now you might have had it in a slightly different form. You might have had it as 9/2 Pi, or you might've had it as a decimal, which would be 14.1 to one decimal place. Yeah. 14.137, et cetera. Okay. So really well done if you've got that correct. Awesome work. Okay. So this is like we did at the start find the area of the shaded region. So the full circle, sorry. The full square has got an area of 12x12, 144. What is the area of my circle? Or what is the radius? The radius is six. So we do 6x6 Pi 36 Pi. And then we do the area of the full square, 144-36 Pi. I'm going to leave it in exact form. Okay. So area of the shape, subtract the area of the circle. Okay. So your turn. Pause the video and have a go in three, two, one. Okay. Welcome back. Now, what is the area of this rectangle? Well, I can see that that is 10. What is this length here? This might be a bit of a sticking point for some of viewers. With the diameter going in that direction, is the same as the diameter going that way. No matter where we take our diameter, the diameter is the same. So that means that that length there is five. Okay. So that length there is five. So here, that means we've got an area of 50 for the rectangle. We need to subtract our two circles. So, what is the radius of this circle? Well, it's 2.5. So we need to do 2.5X2.5 Pi. It's not very good 2.5 is it? That will give us 6.25 Pi. Now I've got two of them, don't I? So the two of them, gives me an area of 12.5 Pi. Okay. So we've got an area of 12.5 Pi. So we need to do 50 subtract that area, to get the area of the shaded region, which you can leave at exact form like that. Or you could write 12.5 is 25/2. So that gives us an area of 10.7, to one decimal place. Okay. So really well done if you've got that correct. Okay. Find the area of this shape. Hmm. What are you thinking? What lengths can you work out? Well, can you see how we have some shape? It might be a rectangle, it might be a square. We'll have to work that out. And we have a quarter circle. So that is length 12. So the radius of this quarter circle is half 12, which means that is also length 12. So if that's 12, then that must be 12 as well. So we actually have a square. So we need to work out the area of the square. And then this time not subtract. We need to add, cause it's two shapes put together rather than a shape being taken away from another shape. So what is the area of this quarter circle? Well, the radius is 12. So 12X12 Pi 144 Pi. And what's the area of the square? The area of the square is 12X12, which is 144. So we have 144 + 144 Pi. And that is our area and exact form. Okay. Your turn. So, it may help you to think of it splitting up like this. So pause the video and have a go. Pause at three, two, one. Okay. Welcome back. Now. Hopefully, you recognise that this is a rectangle of 30 by 12. So to do that, you do 30X12 and you'd get 360 centimetre squared. I didn't put any units on here, did I? That should have been centimetre squared for area. And in fact, I'm not sure I've put in any units on any of the questions, which is very naughty of me. So I do apologise for that. Okay. So that is that area. And then what about this area? Well, the diameter... can you see how the diameter is 30? So that means that the radius is 15. So we have 15X15 Pi, which gives us that, 225 Pi, but then it's half the circle. So you have to half that area. So that's 112.5 Pi. So our total area is 360+112.5 Pi. And if you want, you can write that as a decimal, you can write that as 225/2 like this. Or you could have worked out as a final decimal, but I just chose to leave it as exact form. Okay. So now for a slightly tricky one, and here I have, well, what two shapes do I have there? I have a trapezium here, and I have a sector here. Okay. Now to work out this area of this trapezium, I need the average of the parallel sides, or the mean of the parallel sides, and they are the parallel sides there. And then it's times that by my height. Hmm. What is my height? My height is six, because that length is the same as that length. So because they are both radius, or radii, that length and that length are the same. So that means the height of my trapezium is six, which means I can do the average of the parallel sides, but you might be able to see the average of eight and six is seven, or the mean should I say, times by the height. So that's seven, 14/2= 7 X6, 7X6= 42. Okay. So that is 42 centimetres squared that part. Now, what is the area of this? Well, if it was a full circle of radius six, it'd be 6x6 Pi, which gives us 36 Pi. But, it's only... what, fraction of the circle is it? Well, it's 120 out of 360. How many 120s go into 360? Three. So is that a third of the circle. That is a third of a circle. So a 1/3 of 36 equals 12 Pi. So my total area is equal to 42+12 Pi centimetres squared. Okay. so your turn. Pause the video and have a go at this one. Pause in three, two, one. Okay. Welcome back. Hopefully you had a go. Now, hopefully you found out that this parallelogram has an area of 5x8, which is 40. In fact, it's not a parallelogram, its a rhombus. Because that length is the same as that length. Okay. So that length has got, that has got an area of 40 centimetres squared. Now we need to work out the area of this sector or what fraction of the circle is it? And what would the area of the full circle be? 8x8 Pi, 64 Pi. So we work out that and I think this will give us not a very nice number in terms of Pi. Oh, 40 Pi. Oh, lovely. Why is it such a nice number? Well, it's... I guess because it's half a circle and then 45 is 1/8 of the circle. So I guess it's 5/8 of the circle. Yeah. And that's the reason why. So that's why we get such a nice number there. Okay. So the area of the sector is 45 Pi, sorry, 40 Pi. So we do 40 Pi+40. And that gives that combined area. Okay. So now it's time for the independent task. So pause the video to complete your task and resume once you've finished. Okay. Welcome back. Here are answers. You may need to pause the video to mark your work. Okay. So now it's time for the explore task. In which picture is the biggest area shaded? So pause the video is to complete your task, resume once you've finished. Okay. Welcome back now. Here's what I worked out. I worked out what percentage is shaded in each of them. For, this one, if it's a circle of radius five, one, two, three, four, five, that means that the full circle has an area of 25 Pi. And this is... what fraction of the circle is this? Well, that's into three, six, nine, 12 pieces. So that is 4/12 or 1/3. So 1/3 of 25. Well, so we don't even need to actually work out what that area is. We just can see that that is a third. Okay. So that is 33.3% of the circle. Now this one. Well, that's got a circle of radius three, so that smaller circle has an area of 9 Pi. And the bigger circle has an area of 25 Pi. So what is 9/25? I'll write the pies in, or they'll simplify to one. What is that as a percentage? Well, that's about 36%. How about this one? Well, the big circle is 25 Pi subtract the smaller circle, which is 9 Pi. That's got a small circle, has got radius of 5. So that gives us the same fraction, 9Pi/25Pi, as that one here. Which, you know, if you look at them, this one definitely doesn't look like it's... well, to me anyway, this one definitely doesn't look like the smallest. I'd say this one probably looks like the biggest. And then maybe this one, and then this one. It's interesting how our eyes can kind of deceive us. And that those two are actually the same. I think that's quite interesting, and that this one's less than the both. And so hopefully, you worked that out. Maybe now you could have a go, and see what other shapes you can make, okay? What other fraction of the circle you can shade. Can you make one of these yourself? So thank you very much for all your hard work, over this unit. And I hope you've enjoyed it. And I hope you've learned a lot. If you'd like to, please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak. Thank you very much for all your hard work. Goodbye.

Compound area problems

Compound area problems

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