WEBVTT
00:00:01.660 --> 00:00:05.300
In this video, we’re gonna be talking about how to find the volume of a cylinder.
00:00:06.220 --> 00:00:09.840
First, we’re gonna take a look at prisms and how to work out the volume of a prism.
00:00:10.140 --> 00:00:13.630
And then, we’re gonna explain how a cylinder is a circular prism.
00:00:14.750 --> 00:00:19.390
Finally, we’ll be looking at a few examples of cylinders and how to work out their volumes.
00:00:21.450 --> 00:00:24.480
Before we talk about cylinders then, let’s think about prisms.
00:00:25.360 --> 00:00:28.750
A prism is a 3D shape with a constant cross section.
00:00:29.550 --> 00:00:31.270
For example, here’s a cuboid.
00:00:31.490 --> 00:00:34.490
I’ve marked in the cross section with this blue stripy bit here.
00:00:34.700 --> 00:00:49.870
And if I was to cut this prism at any point, this cuboid, at any point along here, so let’s say through here like this, and look at the slice that I get, I would still have exactly this same cross section.
00:00:51.100 --> 00:00:53.570
Here’s another example of a prism, a star-shaped prism.
00:00:53.800 --> 00:00:58.800
That cross section, which is a star shape, is the same all the way through the length of the prism.
00:00:59.950 --> 00:01:01.320
And here’s a circular prism.
00:01:01.570 --> 00:01:05.170
That circular shape is the same all the way through the length of the prism.
00:01:05.400 --> 00:01:08.410
In fact, that circular prism’s got the special name of a cylinder.
00:01:11.670 --> 00:01:17.780
Now, before we go too far thinking about volumes, let’s talk about a cube with each side of length one unit.
00:01:18.700 --> 00:01:25.350
The cross-sectional area of that cube will be one unit by one unit, which is one unit squared.
00:01:26.760 --> 00:01:36.060
Now, we can work out the volume by multiplying the cross-sectional area by the length, or in this case the height of the prism.
00:01:37.200 --> 00:01:40.830
So, that’ll be one times one, which is equal to one.
00:01:41.050 --> 00:01:43.850
And because it’s volume, it’s units cubed.
00:01:45.410 --> 00:01:53.950
Now, if we take our one-unit-cubed cube and pile it on top of another identical one, we’ll have two cubic units.
00:01:55.030 --> 00:01:57.320
Now, a third makes three cubic units.
00:01:58.350 --> 00:02:00.800
And a fourth makes four cubic units, and so on.
00:02:02.280 --> 00:02:06.250
But what if we started off with two of these cubic units next to each other?
00:02:07.360 --> 00:02:11.280
Now, each time we add an extra layer, we’re adding another two cubic units.
00:02:12.150 --> 00:02:15.000
So, three layers gives us six cubic units.
00:02:15.870 --> 00:02:18.590
And four layers gives us eight cubic units.
00:02:19.240 --> 00:02:29.270
So, the general idea is that for volume you’re taking the cross-sectional area that we had here and multiplying it by the number of layers, or the length, or the height of that prism.
00:02:30.920 --> 00:02:36.110
Now, as we said before, a cylinder is just a prism with a circular cross-sectional area.
00:02:36.920 --> 00:02:44.160
So again, to work out the volume, we just work out the cross-sectional area and multiply it by the height.
00:02:45.400 --> 00:02:48.680
The taller it gets, the greater the volume.
00:02:50.680 --> 00:02:55.880
Now, remember, to work out the area of a circle, it’s 𝜋 times the square of the radius.
00:02:56.710 --> 00:03:01.600
So, if we call our radius 𝑟, the area is equal to 𝜋 times 𝑟 squared.
00:03:02.690 --> 00:03:16.410
And if I let the height of my cylinder, or the length of my cylinder, be ℎ, because the volume is equal to the cross-sectional area times the height, we can say that the volume is 𝜋𝑟 squared times ℎ.
00:03:18.370 --> 00:03:22.150
And that’s the result that we’re gonna be using in our examples in the rest of this video.
00:03:23.610 --> 00:03:27.230
For example, find the volume of the cylinder rounded to the nearest tenth.
00:03:28.350 --> 00:03:32.710
And the circle on the end of our cylinder has a radius of 4.2 feet.
00:03:32.960 --> 00:03:35.820
And the cylinder’s got a height of 6.5 feet.
00:03:36.620 --> 00:03:44.060
So, we’ll mark up 𝑟, the radius, is equal to 4.2 and ℎ, the height, is equal to 6.5.
00:03:45.030 --> 00:03:49.940
So, our approach is gonna be that the volume is equal to the cross-sectional area times the height.
00:03:50.900 --> 00:03:56.720
And since the cross-sectional area is a circle, the area is gonna be 𝜋 times the radius squared.
00:03:57.810 --> 00:04:00.230
So, that’s 𝜋 times 4.2 squared.
00:04:00.490 --> 00:04:04.840
Now, it’s important to remember that it’s only the 4.2 that is squared, not the 𝜋.
00:04:05.870 --> 00:04:15.890
So, that’s gonna be 𝜋 times 17.64, which gives us an area of 55.41769441 and so on square feet.
00:04:16.450 --> 00:04:20.300
But to work out the volume, remember, we do need to multiply by the height as well.
00:04:21.130 --> 00:04:23.300
So, let’s add that to our working out.
00:04:24.490 --> 00:04:36.530
And 55.41769441 times 6.5 gives us 360.2150137, so on, so on, so on cubic feet.
00:04:37.440 --> 00:04:41.330
But the question asked us to round our answer to the nearest tenth.
00:04:42.370 --> 00:04:49.800
So, I’m gonna cover everything up after the tenth and then just do a sneaky peek at the next digit to see whether I need to round that to up or not.
00:04:50.930 --> 00:04:52.950
Well, the next digit is only a one.
00:04:53.290 --> 00:04:58.220
And if it was five or above, then we’d be rounding the two up to a three.
00:04:58.310 --> 00:04:59.580
But it’s not; it’s only a one.
00:04:59.580 --> 00:05:00.980
So, we’re gonna keep it as a two.
00:05:00.980 --> 00:05:07.760
So, our answer to the nearest tenth is 360.2 cubic feet.
00:05:11.500 --> 00:05:13.110
Now, let’s look at a similar example.
00:05:13.110 --> 00:05:18.530
But this time we’ve been given the diameter of the cylinder rather than the radius.
00:05:19.740 --> 00:05:22.510
Now, remember, the radius is half of the diameter.
00:05:23.390 --> 00:05:28.180
So, to work out the radius, we just need to divide 14 by two, or multiply it by a half.
00:05:28.180 --> 00:05:30.090
And that gives us seven inches.
00:05:31.190 --> 00:05:35.570
And the formula for our volume is 𝑉 equals 𝜋𝑟 squared ℎ.
00:05:36.810 --> 00:05:45.390
And so, substituting in the numbers for the radius of seven inches and the height of 13 inches, we’ve got 𝜋 times seven squared times 13.
00:05:45.830 --> 00:05:50.660
Again, it’s important to remember that it’s only the seven that’s squared and not the 𝜋.
00:05:52.030 --> 00:05:55.030
So, that’s 𝜋 times 49 times 13.
00:05:55.840 --> 00:06:02.720
And when we put that into our calculator and round to the nearest tenth, we get 2001.2 cubic inches.
00:06:05.170 --> 00:06:12.320
In this example, we’ve been asked to find the volume of a cylinder with a radius of four centimetres and a height of 14 centimetres.
00:06:12.780 --> 00:06:15.870
We’re also told that we’ve got to leave the answer in terms of 𝜋.
00:06:16.820 --> 00:06:17.900
Now, there are a couple of things here.
00:06:17.900 --> 00:06:19.620
One, we haven’t been given a diagram.
00:06:19.880 --> 00:06:27.380
And two, we’ve got to leave our answer in terms of 𝜋, so this isn’t a matter of just punching the number into a calculator and doing any rounding.
00:06:28.760 --> 00:06:34.340
Now, you don’t need a diagram, but very often drawing a diagram helps you to organize your thoughts about a question.
00:06:34.340 --> 00:06:36.500
So, I would recommend actually doing a quick sketch.
00:06:36.820 --> 00:06:37.950
So, there’s our cylinder.
00:06:38.200 --> 00:06:42.100
It’s got a height of 14 centimetres and a radius of four centimetres.
00:06:43.090 --> 00:06:45.910
Next, we can write out the formula for the volume.
00:06:45.980 --> 00:06:50.300
The volume of a cylinder is 𝜋 times the radius squared times its height.
00:06:52.130 --> 00:07:01.490
And we can substitute in the numbers we’ve been given, so 𝜋 is equal to four squared times 14, which is 𝜋 times 16 times 14.
00:07:02.270 --> 00:07:05.310
And 16 times 14 is 224.
00:07:05.310 --> 00:07:07.820
So, our answer is 224 times 𝜋.
00:07:08.850 --> 00:07:12.600
Now, from the question, both of our measurements were given in centimetres.
00:07:12.600 --> 00:07:15.290
So, the volume is gonna be in cubic centimetres.
00:07:15.970 --> 00:07:16.640
So, there we have it.
00:07:16.640 --> 00:07:17.290
That’s our answer.
00:07:17.340 --> 00:07:20.210
224 𝜋 cubic centimetres.
00:07:21.410 --> 00:07:27.270
So, when the question says leave your answer in terms of 𝜋, it means express it as a multiple of 𝜋.
00:07:29.330 --> 00:07:34.340
Now, we can make things a little bit more difficult by turning these things into word or story problems.
00:07:34.600 --> 00:07:46.230
So, rather than just explicitly saying that we’ve got a cylinder and telling you what the radius and the height are and just doing that calculation, you have to work out the meaning of the different variables from the context of the question.
00:07:46.300 --> 00:07:48.080
So, let’s have a look at some examples like that.
00:07:49.970 --> 00:08:01.870
Given that approximately 7.5 gallons of water can fill one cubic foot, about how many whole gallons of water would be in a cylindrical water tank with diameter 20 feet and height 12 feet, if it was full?
00:08:03.420 --> 00:08:05.560
Okay, first, let’s do a little diagram.
00:08:06.710 --> 00:08:13.960
Here, we have our cylindrical tank completely full of water, depth, or height, of 12 feet and diameter of 20 feet.
00:08:15.050 --> 00:08:20.360
So, first, we can write down that the volume is equal to 𝜋 times the radius squared times the height.
00:08:21.620 --> 00:08:23.440
Now, we can plug in the numbers that we know.
00:08:23.440 --> 00:08:28.270
Well, the radius is half of the diameter, so half of 20 is 10.
00:08:28.560 --> 00:08:31.020
So, the radius squared is going to be 10 squared.
00:08:31.050 --> 00:08:34.950
And it’s important to remember that it’s just the 10 that’s squared, not the 𝜋 as well.
00:08:35.560 --> 00:08:39.370
And the height is 12, so we’ve got to multiply that answer by 12.
00:08:40.770 --> 00:08:50.820
So, this calculation is 𝜋 times 10 squared, which is 100, times 12, so 𝜋 times 1200, or 1200𝜋 cubic feet.
00:08:52.100 --> 00:08:56.460
Now, for the moment, I’m gonna leave my answer in terms of 𝜋 for maximum accuracy.
00:08:56.780 --> 00:09:04.020
If I started rounding to a few decimal places, I would carry these rounding errors through my calculation and my final answer might be quite incorrect.
00:09:05.470 --> 00:09:15.580
Now, we’ve worked out the volume of the tank in cubic feet, but the question says how many whole gallons of water would be in the cylindrical water tank.
00:09:16.540 --> 00:09:21.740
Now, each one cubic foot contains 7.5 gallons of water.
00:09:22.880 --> 00:09:29.070
So, if there are 1200𝜋 cubic feet, there are gonna be 7.5 times as many gallons of water.
00:09:30.490 --> 00:09:38.180
So, the calculation we need to do to work out the number of gallons is 7.5 times 1200𝜋, which I can do on my calculator.
00:09:39.010 --> 00:09:41.650
Now, it’s okay to round right at the end of the question.
00:09:42.020 --> 00:09:46.570
And the question said about how many whole gallons, so I need to round to the nearest whole gallon.
00:09:46.570 --> 00:09:51.250
So looking at our number here, that’s gonna be 28274.
00:09:52.530 --> 00:09:58.950
So, we can write our answer out nice and neatly at the end, 28274 gallons of water.
00:10:01.770 --> 00:10:11.040
Which has the greater volume, a cube whose edges are four centimetres long or a cylinder with a radius of three centimetres and a height of eight centimetres?
00:10:12.280 --> 00:10:19.760
So, what we’ve got to do here is calculate the volume of the cube and also calculate the volume of the cylinder and then compare the two.
00:10:21.200 --> 00:10:26.790
So, first, the cube, let’s draw a sketch, four centimetres by four centimetres by four centimetres.
00:10:27.010 --> 00:10:30.230
And the volume is just gonna be four times four times four.
00:10:31.430 --> 00:10:36.700
And since the length units were centimetres, our volume is gonna be in cubic centimetres.
00:10:37.010 --> 00:10:39.900
And four times four times four is 64.
00:10:40.130 --> 00:10:43.270
So, the volume of the cube is 64 cubic centimetres.
00:10:44.470 --> 00:10:46.420
Now, a quick sketch of the cylinder.
00:10:47.550 --> 00:10:52.570
And use the formula the volume is equal to 𝜋 times the square of the radius times the height.
00:10:54.390 --> 00:10:58.310
Now, let’s remember that that squared only applies to the three.
00:10:58.340 --> 00:10:59.730
It doesn’t apply to the 𝜋.
00:11:00.000 --> 00:11:02.550
So, we’ve got 𝜋 times three squared times eight.
00:11:03.780 --> 00:11:06.160
And three squared is nine.
00:11:06.160 --> 00:11:07.920
So, nine times eight is 72.
00:11:07.920 --> 00:11:09.870
So, we’ve got 𝜋 times 72.
00:11:11.230 --> 00:11:13.570
Now, it doesn’t ask for a level of accuracy in the question.
00:11:13.570 --> 00:11:19.510
But I’ve rounded that to two decimal places to give me 226.19 cubic centimetres.
00:11:19.850 --> 00:11:29.070
So, again, the measurements were in centimetres, the volume is in cubic centimetres, and the two numbers that we’ve got to compare are both in the same units, cubic centimetres.
00:11:30.370 --> 00:11:31.700
Now, we can compare those.
00:11:31.970 --> 00:11:39.730
And 226.19 is clearly a lot larger than 64, so the cylinder has got the greater volume.
00:11:42.520 --> 00:11:44.420
Now, let’s summarize what we’ve learned.
00:11:44.710 --> 00:11:48.900
First, a cylinder is a type of prism with a circular cross section.
00:11:50.660 --> 00:11:58.890
Next, to calculate the volume of a prism, you find the area of the cross section and multiply that by the length, or sometimes called the height, of the prism.
00:12:00.330 --> 00:12:06.000
The volume of a cylinder, 𝑉, is equal to 𝜋 times the square of the radius times the height.
00:12:07.760 --> 00:12:14.300
And a top tip, always check were you given the diameter or the radius of the cylinder in the question.
00:12:14.470 --> 00:12:15.500
That’s really important.
00:12:16.870 --> 00:12:24.530
And finally, when answering story problems, make sure to read the question carefully to find the relevant information and check your units.
00:12:25.490 --> 00:12:31.020
Also, always consider drawing a diagram cause it can be really helpful to organize your thoughts on the problem.