00:00Hi everybody. So in this
00:02lesson we're going to look
00:03at volume and surface area
00:04of 3D shapes. Now you'll
00:07see from the formula booklet
00:08they give you loads of
00:10different formulae and they're in
00:11two different places. One, well
00:13if you're not familiar with
00:14the prior learning section of
00:16the formula, if you check
00:17it out, they give you
00:17all these formulae. And then
00:19in section 3 .1 they
00:21give you the volumes and
00:23surface areas of right pyramids,
00:25right cones, surface areas cones,
00:28volume of spheres and surface
00:30area of spheres. So, well
00:33actually while I'm here, a
00:34right pyramid and a right
00:36cone, all that means is
00:37that the base is a
00:39regular polygon, like a regular
00:42triangle or square. And the
00:48apex, like the pointy bit
00:49at the top of the
00:50perimeter of the cone is
00:51directly above the center of
00:53the base, and that actually
00:55is quite important.
00:56So to give you them
00:57and then you get all
00:58the other kind of basic
00:59ones here like the circle
01:00and cuboid and cylinder and
01:02prism. Okay, so you might
01:06think this is a fairly
01:07straightforward topic and it actually
01:11is but the questions that
01:12the topic is straightforward but
01:13the questions they can give
01:15you are well they can
01:16be very very challenging as
01:18you're going to see when
01:19I do these two these
01:20two questions here. I've taken
01:22them from two examples I've
01:23taken them
01:24from past papers. So let's
01:27go through them. So, well,
01:28before I do it, when
01:31obviously when you're given a
01:33formula, you just need to
01:34find the, find the, the
01:36radius or the height or
01:37whatever it wants and sub
01:38it into the formula and
01:39that way you can get
01:40your area of the volume.
01:41However, they can easily make
01:43the question more challenging by
01:45giving you the volume and
01:46then saying find the radius.
01:47So you have to rearrange
01:49our, that's the very least,
01:50use your calculator to
01:52figure out what's going on.
01:54Okay, so let's go straight
01:56into example one. Okay, example
02:02one. The height of the
02:07cuboid is equal to the
02:08height of the hemisphere. So
02:10we've got a cuboid here
02:12on a hemisphere on top
02:14of it. And it says
02:17the height of the hemisphere
02:18is equal to the height
02:19of the cuboid
02:20write down the value of
02:21x. Now I should say
02:22a hemisphere is half a
02:25sphere. So the height of
02:27a hemisphere is actually going
02:29to equal the height of
02:30the hemisphere will equal the
02:32radius of the full sphere.
02:34Because that's what hemispheres is.
02:35It's half of it. So
02:36the height of this is
02:37actually equal to the radius.
02:39I might even put that
02:40as r. That's going to
02:41equal the height of this
02:42is going to equal the
02:43radius. Now, this probably left
02:46out a few details. It
02:48says
02:48as in the question I'm
02:49sure that it's exactly on,
02:52so the hemisphere is exactly
02:53on the edge of this
02:54cuboid here and here and
02:56here. So the diameter of
02:59this hemisphere has to be
03:02six, because it goes from
03:03here to here. So the
03:04diameter of six, which means
03:05the radius equals three, which
03:08means this is equal to
03:09x, which means x equals
03:10three. So a, x equals
03:12three. It's a write down
03:13question, which means all you
03:15have to do is just
03:15write down the answer.
03:16Okay, fine. Part B is
03:19the more challenging bit. It
03:20says calculate the total surface
03:23area. Okay, the surface area
03:28equals, right? There's a lot
03:30of different things going on
03:31here. Firstly, this, let's say
03:37this front kind of section
03:38here. This is a rectangle
03:40and it's just going to
03:42be six times three. So
03:43in brackets, I'm going to
03:44But six times three. But
03:48there's four of them. There's
03:49one here, one here, one
03:51at the back, and one
03:52at the side. So this
03:53is six times three times
03:56four. Six times three times
04:01four, because there's four of
04:02them. Then at the bottom,
04:05there is a square. Six
04:08times six. Six times six,
04:11I put that in a
04:12bracket.
04:12Now the tricky bit is
04:15this thing here. There's like
04:18a this part here and
04:22this trying to get how
04:25are we going to get
04:26this area here. Well, let's
04:33let me show you something.
04:35I am going to draw
04:37a circle. Here's a circle.
04:40This is the hemisphere. If
04:45you like, let's imagine we're
04:46looking down on this. So
04:47if you were looking down,
04:49you'd see a circle and
04:50you'd also see a rectangle.
04:55You would see, let me
04:56make this a bit bigger.
04:59Okay, you would see that.
05:04And if I shaded in
05:06red, this is what we're
05:08looking for.
05:08kind of area here. This
05:10red bit plus this red
05:12bit, plus this red bit,
05:17whatever, plus this red bit.
05:19Okay, how do I find
05:20the area of that red
05:21bit? Well, I'm going to
05:22do the square minus the
05:25circle. The square minus the
05:27circle, that would be, so
05:29I'm going to add, the
05:30square is 6 times 6,
05:326 times 6, minus the
05:35circle is
05:36pi or squared if the
05:38area of a circle is
05:39pi or squared. Surely we
05:41know that. Surely we've heard
05:43that before, but if we
05:44don't, there it is. They
05:45even give you the area
05:46of the circle. So it's
05:46pi or squared, which is
05:48pi three squared, because I
05:50know the radius is equal
05:51to three. And then finally,
05:56I'm not done, I need
05:58to add the area, because
06:02he wants the surface area,
06:04I need the surface area.
06:04the surface area of the
06:06hemisphere. Now they give me
06:08the surface area of a
06:09sphere. It's 4 pi or
06:12squared, so it's 4 pi
06:15or squared, 3 squared. But
06:19it's not a full sphere,
06:21it's a hemisphere, so I
06:22need to have this. So
06:24I'll just multiply by half.
06:27Okay, look at that. That
06:28is a long equation.
06:33I'm just gonna use my
06:34calculator. There's no way I'm
06:36gonna try and figure that
06:36out in my head. Let
06:38me move this up here.
06:40I'm gonna type in all
06:42of this. It's six times
06:44three times four plus six
06:51times six plus six times
06:54six again. Or I can
06:56just say, let's do, let's
06:57do exactly how we see
06:59it. So it's plus six
07:01times six again minus I
07:05want to pi or squared.
07:07I'm going to say, well,
07:09let's use the pi button.
07:11It's pi times or squared
07:14times three squared, pi times
07:18three squared plus four times
07:25plus four times pi times
07:29times 3 squared again, times
07:313 squared, and then I
07:34need to do times a
07:36half times, let's just do
07:380 .5. Okay, is that
07:42the same thing? Six times
07:446 plus 6 and 6
07:45minus pi times 3 squared
07:47plus 4 pi 3 squared
07:49times a half. Yes, it
07:50is. Press Enter and I
07:52get 172 .274, 172,
07:57point two seven four. It's
08:00surface area so this would
08:02be well if this was
08:04sent to me this this
08:04would be sent to me
08:05the squared if it was
08:06meters obviously meter squared or
08:08let's just say unit squared.
08:10It doesn't ask me this
08:11question but if it did
08:12make sure we put down
08:14the right answer. Okay fine
08:17that's the that is the
08:20first example. The second example
08:22gives me an even strange
08:25looking shape. It's as calculate
08:27the volume of this shape.
08:29Now this is a cone
08:30which has had another cone
08:33taken out of it. So
08:35first thing I want to
08:37know is the volume of
08:39a cone formula. Do they
08:42give us the volume of
08:43a cone formula in the
08:44formula booklet? Yes, they do.
08:47It is here the volume
08:48of a right cone. It's
08:49a third pi r squared
08:52h where
08:53H is the height. The
08:57third pi r squared H,
09:00so it's equal to a
09:01third pi r squared H.
09:09Okay, now the volume of
09:12this, the total volume, total
09:15volume, is equal to the
09:18volume of the big cone,
09:21the volume
09:21with a big cone minus,
09:24so I'm going to do
09:25volume big minus volume small.
09:30And when I say small,
09:31this small cone here. Now,
09:36to get the volume, I'm
09:38going to need the radius
09:40and I'm going to need
09:41the height, right? So I'm
09:43actually going to make two
09:45little drawings here. So let's
09:48go down here.
09:49The big cone, well actually
09:52let's just draw it on
09:54here. The big cone comes
09:56down here and across here.
09:58So I have a triangle.
10:00This is a right angle
10:01and this is 21 and
10:05this is 35. So if
10:07I want to find this
10:08big height, let's call it
10:10x. I can use Pythagoras'
10:12theorem and note how often
10:13Pythagoras' theorem comes up. So
10:17in order to find
10:17And the volume, I need
10:19this height, this big height,
10:21and I don't have it.
10:23So I'm gonna do x
10:25squared plus 21 squared equals
10:3235 squared. That's my a
10:33squared plus b squared equals
10:34c squared. x squared equals
10:3735 squared minus 21 squared.
10:41And then x equals the
10:43square root of that.
10:45So it's the square root
10:48of 35 squared minus 21
10:56squared and gives me 28
10:59exactly. Nice. So that height
11:02is 28. What I don't
11:07have in the small cone
11:09is the radius. Now let
11:12me draw another
11:13Here, and I'll just draw
11:17this red line up here
11:19like this. Let's call this,
11:22and I'm gonna call it
11:22r, because that will confuse
11:24me. We've got two different
11:26r's. I'm gonna call it
11:27y. So again, Pythagoras, I
11:31can say y squared, y
11:35squared plus 12 squared equals
11:3815 squared, y squared equals
11:4112 squared.
11:41equals 15 squared minus 12
11:45squared. I'm going to do
11:46this quickly because I can
11:47actually do this in my
11:48head. This is going to
11:49be 225 minus 144. This
11:53minus 144 is 81. And
11:56the square root of 81
11:57is 9. So now I
12:00have all the pieces in
12:02the puzzle. I can say
12:04the volume of the big
12:06cone is a third times
12:08pi times
12:09or squared, my r is
12:1231 pi times 31, 21
12:18squared, that's pi or squared
12:20times H, which is X,
12:2428. And I'm gonna subtract
12:26the small cone because look,
12:29that has been, the small
12:30cone has been removed. So
12:32I'm gonna subtract a third
12:35times pi times or squared.
12:37which is y, which is
12:399 squared times pi r
12:43squared times the height of
12:46the small cone, which is
12:4912. And all of this
12:52gives me, let me move
12:54this down to here. And
12:57I get a third. And
13:01let's do a third here.
13:04A third.
13:06Pi times a third. Pi
13:12actually put in the multiplication
13:15sign. There's sometimes it's annoying.
13:17A third times pi times
13:1921 squared times 28, the
13:26height. And then minus, again,
13:28do it all. And we
13:29won't go minus a third.
13:31And we do this again,
13:33a third.
13:34times pi, I'm going to
13:38do times, I'm going to
13:42do, I didn't want to
13:42do that, I'm going to
13:43do pi, a third times
13:45pi times 9 squared times
13:5112. And that gives me
13:5511 ,000, I'm going to
13:58do that go 11 ,912
14:01.9,
14:0211 ,912 .9. If this
14:06was centimeters, then all of
14:08this is centimeters cubed or
14:10meters cubed or whatever the
14:12case may be. Okay, so
14:14clearly you've seen there that
14:17these questions are not easy
14:18by any stretch of the
14:19imagination. But remember, you have
14:25all these formula and it's
14:27just the case of kind
14:28of figuring out what they're
14:30asking for and what pieces
14:31of the puzzle you are
14:33missing and then doing this
14:34to calculations to find it.
14:36I always find drawing little
14:38kind of separate shapes, separate
14:39triangles helps in these kind
14:41of situations.