Learn AISL 3.1.3 Volume and surface area in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 3.1—3d space, volume, angles, midpoints.
Learn AISL 3.1.3 Volume and surface area in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 3.1—3d space, volume, angles, midpoints.
The video lesson focuses on calculating the volume and surface area of various 3D shapes, including cuboids, hemispheres, and cones. It emphasizes the importance of understanding the relevant formulas, which can be found in the formula booklet, and how to apply them in different scenarios, including rearranging equations when given the volume to find the radius.
Key points include:
Additionally, the lesson covers the volume of cones, using the formula , and demonstrates how to find missing dimensions using Pythagorean theorem when necessary.
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
00:23surface 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.