The volume of a 3D shape is the space that occupies the inside. The surface area of a 3D shape is the total area of all the faces.
To calculate the volume of an object depends upon the shape of the object.
Cuboid volume = length x width x height
Prism volume = area of cross section x length
Cylinder volume = pie x radius2 x height
Hello, everybody. We're going to be looking at three dimensional shapes, looking at volume and surface area of a selection of three dimensional shapes. This, of course, is extremely important in all forms of construction going back thousands of years to ancient buildings like the Great Pyramids, Great Wall of China. Skyscrapers today that we see are basically just giant cuboids. If you're going to see how much glass you need for the windows, that will be the surface area.
If you're trying to see how much space you can fit in container, that's a volume. So, yeah, it's a very useful thing that goes back a long way, but we're going to be looking at how to calculate these things today. So volume, of course, is the amount of space inside, for example, how much water you could use to fill up a tank. And the volume for a simple cue board is calculated by doing the length times the width times the height LWH. So in this case, we've got ten times two, times three, which is just going to be 60 centimetre cubed.
As volume is measured in cubes, whether it's centimetre cubes or metre cubes and so on, surface area is always more complicated than volume. And here's the formula for the surface area. It's two times everything in the bracket, where the bracket is the length times the width. So that would be ten times three, which would be this blue bit, the width times the height, that's three times two, which is this red one, and the length ten times the height two, which is this yellow section in the front here. So we've got two times each of these three faces that we see.
Why two times each of the three faces that we see? Well, you need to count up the two dimensional surfaces, the two dimensional areas on the shape. For example, if you had a present that you're going to wrap for your family at Christmas, the amount of wrapping paper that you wrap the box in is a two dimensional surface. So we've got the top, we've got the right and we've got the front. But what we've forgotten is that there's two lots of those, because there's also the left, which is the same as the red one.
There's the bottom, which is the same as the blue one, and there's the back, which is the same as that front yellow one. So you've got the top, bottom, right, left, front and back, which is where this two comes in. So essentially, all you're doing is with this Lwlh is you're getting these rectangular areas of the red, blue and yellow faces and then you're doubling them for the back, the bottom and the left. So let's see what happens there. We've got two times length times width would be 30, ten times three.
Then w width times height is six, and then length times height is going to be 20. So we've got two lots of 30 plus six plus 20 is 56. And so the final answer is 112 centimetres squared is the surface area of this cuboid.
Looking now into the volume of this prism. The volume of the prism is always volume is the area. Let me go back and do that again also here.
Okay, we're now going to be looking at the volume of the prism. So the prism is any shape where you've got a cross section. In this case that would be this triangular face that's pulled along a length and has the same cross section throughout the whole shape. A cylinder, by the way, is another example of a prism because a cylinder is a circle hold up by the height. So a cylinder is another example of a prism.
So what we do here is we get the area of the base. It's that base area times the length, which in this case is ten into the page. So the volume of this prisoner is the area of this base. Area of the triangle is a half, five times four times that length, ten. So we end up with ten, which is the area of this base times by ten the area on the length.
And we get the volume of this prism to be 100 centimetre cube. If you're going to do the surface area of a prism, but more complicated, you need to see that there are five faces. You've got the front and back triangle, which we just calculate the area was ten. Then you've got this back rectangle here, which would be 40. This bottom rectangle on the base here, five times ten is 50.
So you got 1234. And then you've got this other rectangles. You've got five faces in total. And then you would need Pythagoras' theorem to get the length of this diagonal. And then you would do length times width.
So surface area is always more complicated in the volume, especially in the case of a prism. You'd be Pythonaggress's theorem unless you were given that slope.
Okay, volume of a cylinder is pi R squared, which is the area of this circle at the bottom times by the height. H in this case is five. We have a radius of four and a height of five. So the volume is just this base times the height. So it's pi R squared.
R is four. So we've got 16 times pi. Just going to leave that as pi rather than do 3.14 times by the height times by five. And so you get 80 pi centimetre cubed for the volume of this cylinder. Sometimes questions will ask give your answer in terms of pi and that's how you would do it.
You would just leave the pi symbol as it is. Now, moving on to surface area of a cylinder, a lot more difficult.
So the surface area of a cylinder, just to show you the formula to begin with, is two pi R squared plus two pi RH, where on earth does all that come from? Well, the two pi R squared bit is easy enough to understand. The area of a circle is pi R squared. And so we've got this circle up here making the top, and we've got this circle down here forming the bottom. And so the two pi squared is just the two areas of the top and the bottom.
But what's the surface area of the curve bit around left and right? Well, if you imagine a tin of beans with a label on it and then you peel the label off, and if you haven't done this before, give it a shot, you will find it is a rectangle. A lot of people find that difficult to understand. But if you unco the curved surface around this cylinder, it will curl out to a rectangle. The height of the cylinder will be the height of this rectangle here.
And the length all the way around is the length all the way around the cylinder, which is our circumference of the circle, two pi R. So that's where it comes from. So if I'm going to calculate the surface area of this cylinder, I first of all do the two circles, top and bottom, which is two pi R squared, which is two times four times four. Four squared is 1632 pi. That is the area at the top and bottom.
And then I've got two times pi times four times the height of five. And so I've got top and bottom, 32 pi plus 40 pi for the area of the rectangle. And so finally, 72 pi is the centimer squared. 72 pi centimetre squared is the surface area of this cylinder. Again, cylinder is a very difficult one to remember for calculating the surface area.
But if you can picture the net here, that will make it a lot easier, okay, to calculate the volume of any pyramid like. So it is one third the volume of the containing cuboid. So if you imagine here we have a containing cuboid and we're just going to use the same dimensions for the cuboid we had previously, two x three by ten. This cylinder is a third of two times three times ten. It's a third the volume of our cuboid, which was 60 centimer cubed.
So the volume of this pyramid is 20 centimetre cubed.
Similarly to the pyramid, a cone is one third of the volume of the containing cylinder. So again, we're using the same dimensions we had for our cylinder last time, four, a radius of four and height of five. And we're going to find the volume of this cone. So it is just one third pi R squared, h, which is one third, and R squared is 16 times the height five, which is 80. So we've got one third times 80 pi.
And so we've got 80 over three pi centimetre squared. And you can do 80 divided by two in a calculator and get the decimal and you can change pi to 3.14 to get the decimal if you wish, but that's the idea. Cone one third the volume of the containing cylinder. Surface area of a cone. Another difficult one to remember, I'm afraid it's pi R squared, which is that area of the circle on the bottom.
So we've got 16 pi plus pi times the radius times this slope diagonal L. And that gives you the curved area around here. Very difficult to visualise on a net. That one difficult to calculate, but this is the formula anyway. And so you would have pi x four times in this case, using Pythagoras to get this length from the radius of four, height of five 6.4.
You get 6.4 as the length after doing Pythagoras. And so we finally have 16 pi plus 25.6 pi. And the final answer therefore, is 41.6 pi for the surface area of this column.
Final shape to calculate the volume of a sphere, it is four thirds pi times the radius cube, four thirds pi R cubed. So in this case, we have a radius of ten. All spheres of the same shape. It doesn't matter, they're just different in size. So all you ever need is the radius and you know everything you need to know about the sphere.
So we've got four thirds times pi times ten, times ten times ten times ten is cubed, which is 1000. So we've got four thirds times 1000 pi centimetre cubed. And again you can do four times 1000 divided by three to get the decimal and times by pi if you wish. Surface area of this sphere is four pi R squared. Notice that for the volume formulas in terms of our cube, which makes sense, we've got centimetre cubes and for the surface area it's in terms of our squared, which makes sense.
We're dealing with squares since we square for surface area. So we've got four times pi times ten squared. So we have 400 pi centimetre squared for the surface area of that sphere.
Okay, let's have a look at a basic example here.
Okay, let's have a look at a basic example here of how to calculate the surface area of this cylinder. So what we'll need to do is get our formula for the oops.
Okay, let's have a look at the base example here. We're going to calculate the surface area of this cylinder. So we just need the formula for our surface area of the cylinder, which if you recall, was two pi R squared for these two circles plus two pi r H. For the surface area of the curve back around left and right. So we get radius of two.
Two squared is four times two eight. So we have eight pi for the circle on the top and the bottom. And then substituting in for R and H we have two pi times r, which is two times height, which is five. And so we have eight pi plus 1020 pi. And so final answer is 28 pi.
And if the question did ask you leave your answer in terms of pi, this is exactly how you would and if the question did ask you to leave your answer in terms of pi, this is exactly how you would have to leave it. With this pi at the end, you wouldn't convert to a decimal. And I forgot my centimetre squared.
Okay, time for an exam style question here. This is a difficult one. It's the surface area of a cylinder, and it involves algebra. It's got a diameter of two X and a height of X. Very difficult question to answer this.
If you've think you can do it, please post your answers in the comments section, or you can email me directly. My name is Adam, a tutor for Sherpa or any other sherpa tutor will be able to give you the answer if you contact them. Good luck with this one, guys. It's tough.
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