Geo 10.1 Surface Areas of Prisms and Cylinders

It's a Pentagon and it's the same Pentagon as this bottom. In fact, we call prisms by their base shape. So this is a Pentagon base, so it's a ping pentagonal prism, all right? So these bases are connected with these faces and we call them lateral faces. And this is a lateral edge. It connects the two bases together and is one side of a lateral face over here you can see this is a triangular prism because it has triangles as its bases. All right? Now, a couple of things. First thing is sometimes real great when these shapes are like this. Bases on top and bottom. But sometimes we can turn them around here too. All right? The bases could be actually on the sides. Not everything's nice and neat and math. So you have to figure out where the bases are, all right? It's not as hard as I'm probably making it out to be. All right, so let's talk a little bit more about these bases. The altitude of a prism, the altitude of a prism is the perpendicular segment that joins the plains of the bases. It's the height of the prism is the length of the altitude. So this edge here is the height. It is the perpendicular distance from one base to the other. Perpendicular distance from one base to the other. 

Notice when the height is a lateral edge, we call these right prisms. In other words, when they are perfectly straight up and down, all right? They need perpendicularly. We call those right prisms because they have those right angles. But prisons can also be called oblique. I like to think of the Leaning Tower of Pisa when I think of oblique prisms or things like that. It's like baleen, they're just shifted over a little bit, all right? And it's still the height is still going to be the perpendicular distance there. We're going to work mostly with the right prisms, but every now and again, we're going to talk about oblique prisms, all right? All right, so like I said, I write prism is when the lateral faces are rectangles and a lateral edge is the altitude, all right? That edge is actually the altitude. And unless you are told otherwise, always assume for our purposes that any prism is a right prism. I will tell you otherwise if it's oblique. The other way you'll know it's oblique because the lateral faces are non rectangular. Look at this. Every face here is a rectangle. This little one in here. It's a smaller rectangle, but it's still a rectangle. These are not rectangles. Yes, they're quadrilaterals. But they're not rectangles, all right? But again, most of the time right now, we're just going to be dealing with right prisms. 

So let's take a look. What I've done here is we want to find the lateral area. So the lateral area is all these signs. I want to find the area of this side and add it together with the area of this side, all the lateral sides, and add it together with the aerial. This side. And add it together with the area of this side. The only thing I don't care about, I don't want to find the area of this base or this base, all right? And you can see I've kind of unfolded this. What I unfold this, this is the base that we don't want to. We're not looking to find right now. This is the other base that we're looking to not find right now. So if you look at this, this lateral area is right here and what could we describe this lateral areas? Well, it's a rectangle. All right? It's a rectangle. We just unfolded that shape, made a net and made a rectangle. Now this is four, followed by this side is three, followed by this side is four. Followed by this side is three. That's just the perimeter of what? It's the perimeter of the base. All right? So the length of this rectangle is just the perimeter of the base. And the width of our rectangle is right here, it's our altitude. It's our height. So we can find this lateral area by just multiplying the perimeter of the base times the height. And if it sounds like a formula, ladies and gentlemen, it probably is. 

So the lateral area of a right prism is perimeter times height and remember this perimeter is the perimeter of the base. All right. We'll talk more about that in a second. We'll do one. Now let's take a look here. Well, if we have found all this right here, we found the area of all this. We only have two more sides to cover the entire thing. All right? And surface area is the sum of the lateral area and the two bases. So the lateral area, we know how to get and then we're going to add the two bases, so lateral area plus two bases, let's take a look, lateral area plus two bases. That's what it is. Now, check this out. Right now, it's pretty nice. We have two rectangles, right? Two rectangles right here. Could these shapes change? Yes. Notice this, this is lateral area plus the area of two bases. This big B stands for the area of the base. Now we have a lot of different area formulas, all right? We have air of a rectangle. This one is length times width. We have the area of a triangle. We have the area of lots of different shapes. So this formula could change quite a bit. But in general, what we're going to have is our surface area will be the perimeter times the height, which is just our lateral area formula, plus two times the area of the base. And again, that base formula will change depending upon what our figure is. 

All right, well, so let's try one. So find the lateral and surface areas of the following prism. So first thing we need to do find the lateral area. And what is our lateral area of formula? Perimeter of the base times the height. So I need the primer of the base so I have three plus four plus ah man, I don't know what that is. But it's a right triangle, so I know I can use Pythagorean theorem to find it. Now, this one, some of you are jumping out of your seats right now going, I know the answer. I know the answer. And if you do great, 'cause this is one of the all time big ones, right? If you don't remember, you can do a squared plus B squared equals C squared. And then we get 9 plus 16 equals X squared. 25 equals X squared, and then X equals 5. So this side here would be 5. So now we know our perimeter is three plus four is 7 plus 5 is 12. All right, now we need to find the height and again the height connects the two bases together. What is our height in this case? It is 6. So multiply it out 12 times 6 is 72. And we got units here. We got centimeters. 

Now anytime we're in area, what kind of unit do we have? We have squared units, right? So the lateral area of this is 72 square centimeters, all right? So now let's find the surface area. The surface area is the lateral area plus two times the area of the base. What we know the lateral area, that's nice. 72 plus two times the area of the base. What base do we have? We have a triangle. What's the formula for the area of a triangle? One half base times height. Now I know this gets confusing. We got a base here. We got a base here. That's why I kind of did this as a big B big B means area of the base. Now I'm looking at that base and this triangle formula has base times height. Remember our base and height are perpendicular in triangles, so the base here would be three, so this is going to be two times one half times three times four, all right? Plus 72, okay? The next thing about this, you know, two times a half, that's just gone. Three times four is 12. 72 plus 12 is 84. Centimeters squared. All right? So that right there, now we have found the area, a lateral area, and the surface area of a prism. All right? 

Next one. Cylinder, a cylinder is another solid that has two congruent parallel bases that are circles. This time it's very specific. It's not, it's like a prism, except for the bases are circles. All right, so our bases are circles here and here. Again, right cylinders that perpendicular distance is perfectly straight up and down. Oblique are this is exactly like the Leaning Tower of Pisa. It's leaning over a bit, so our perpendicular distance is not exactly that side length right there. All right, so let's find one. Oh, let's take a look at these formulas. Here we go. Now I've kind of unwrapped this. You got to kind of imagine now if you have a hard time imagine this, actually get a toilet paper roll, cut it, and unroll it. You're going to see that what you have is a rectangle. When you unroll that, all right? Now what is this rectangle? Well, this is the area the cut you made, all right? And this here, this two pi R, what is two pi R? Two pi R is circumference. So I'm just unwrapping the circumference of this circle, and that is our width, right? So length times width, if you want to find lateral area right here, it's just going to be two pi R times H and that sounds like a formula, and it is. Two pi R times H that is our lateral area. 

All this right here, our lateral area, two pi R times H, and that's because we rolled that out. All right. Surface area then, we know what the nice thing about surface area with cylinders is you know what the formula for is every time. It's the same, actually, as a prism. Two times B, but we know what the area of B is, we know it's the area of a circle is pi R squared. So our surface area formula, we could even write it out specific every time we could write the length, the lateral area is two pi R H plus, the two times the area of the base, pi R squared. All right? So let's try one of these. Okay, so find the lateral and surface area of the cylinder. Here we go. Let's find the lateral area. So lateral area equals two pi R H so two pi, what is R? What is our radius if our diameters ten are radius has to be 5? And our H connects the two bases together, so that's gotta be 12. Two times 5 is ten, ten times 12 is a 120 pi meters squared. All right. Surface area. So surface area is lateral area plus two times our base area of the base and it's pi R squared. So we know this is a 120 pi. Two pi times our radius again was 5. And 5 squared. So a 120 pi plus two pi times 25. So that gives us a 120 pi. Plus 50 pi, and that is a 170 pi meters squared. All right. 

So that's finding the surface area of the cylinder. All right, so let's try some more examples here. Let's find the surface area of each figure. Now the great thing here, surface area, when we find surface area, we're finding lateral area already in it. So let's do this. This is a one, two, three, four, pentagonal, a regular pentagonal prism. Regular because all the sides are the same. So surface area equals perimeter times height, that's the lateral area, plus two times the area of the base, what's the area of formula for the base of a regular hexagon, one half times the atom times a perimeter, all right? All right, right off the bed. I'm just going to two times a half cancels out. All right, let's just take care of that right now. So, perimeter, I have to add all the sides up, 7, 7, 7, 7, 7. So 35 times my height is 9. Plus, the app of them is 4.8. Times the perimeter. And again, we know it's 35. All right. We multiply this out. We get 315 plus a 168. Add those together and we get 483 inches squared. All right, so you got to remember this formula two. I'll provide you with the formula sheet. I'm not going to tell you what formula to use now. All right? I'll probably say these are area formulas. You need to recognize which ones you have to use. 

All right, so now, round of veneers tenth, so here's tenth, that would be 44 and 83 point zero. All right. Over here, surface area of a cylinder. So we got two pi R H plus two times area of the base, which is pi R squared. All right, so let's plug in what we know. Our radius is for a diameter is 20, our radius is ten. Our height is 8 plus two times pi times our radius again, squared. So two times ten times 8 to a 160 pi. Plus ten squared is a hundred, a hundred times two is 200 pi. Add those again and we get 360 pi. Now remember, this is the exact answer. It's exactly because pi is one of those irrational numbers. We can not we can round and guesstimate with it, but we can not get exact if I plug in 3.14 for it. It's not going to be the exact same thing. So now when I say round of the nearest tenth, I'm going to be using the calculator's approximation. All right? So let's take a look at what I mean. So I want to do 360 pi. So let's go to our calculator. 360 pi. Our pi symbol is right here above the exponent button. So if I have to get to the blue, I have to press second, then I can press that right there, the pi button. 

Now this is going to give me my decimal approximation. I'm using this value of pi. I don't want you to put 360 times 3.14 in. I want you to do it this way. So all of us have the same answer. All right? Because I want to make sure you guys know how to round as well. Here's why. So we went around to the nearest tenth. So I go, here's my tenth. I look before 7 rounds at 9 to what, a zero. Oh, that means I have to round this up to a one. So this is actually going to be 1131.0. All right. 1131.0 and kilometers squared. Holy cow. That's a huge cylinder. Probably not real life right there. Kilometers are really big, right? All right, so pause the video just kidding. Let's watch mister Kelly. Let's do a little application. Sullivan has become tired of all the bullying from mister Kelly during his videos. One day he catches him and starts lowering to a vat of acid. He tells him the only way he consume himself is if he correctly figures out the square feet of paint, so he used to paint the outside of the vat. That is a diameter of 6 feet and his ten feet deep, Kelly guest is 30 ft². Will he survive? Let's take a look. So we want surface area, right? Now surface area is lateral area plus two bases. But check this out. If we're dipping them in, do we have two bases and we're missing a base. 

So ours is really going to be just base. So what's our lateral area to pi R H plus the area of our base is going to be pi R squared. Let's take a look. Our radius, if the diameter is 6 feet, our radius is three, our height is ten plus radius squared is going to be three squared is 9, right? So two times three times ten is 60 pi. Plus 9 pi, so that's going to be 69 pi squared. Now he said 30 feet, this is already 69 times three is way more than 30 feet, so is Kelly's going to survive. Of course not. Dip them in celibate. Boom, that's what you get for all those funny pictures and videos making fun of son of a. All right, now you try some on your own pause the video and work them out. All right, welcome back. So these are kind of good because I think there's a little trick here on the one. Let's find the surface area. So surface area equals two pi RH plus pi R squared. Two pi times the radius is three. Times the height is 8. Plus R squared is three squared. So that's going to be 9. So that's 9 pi. Three times two is 6. 48 pi plus 9 pi. Oh, I forgot this times two. So 18 pi, all right? And 48 plus 18 is going to give us 66 pi. Squared. That would be the exact answer if we wanted to approximate it. 

We plug it in. We are in our calculator and round it up. All right, now this one's tricky. This is a rectangular prism rectangular prisms are tricky because this could be the bases, these two sides, right? Or the front two sides, the front and back could be the bases. Or the top and the bottom. It all really depends. All right? So here's what I'm going to do. I'm going to say these two sides are my bases. If you didn't use those two sides, you should still get the same answer. So let's take a look. If that's 8, this is 8, this is a and 8. All right, so surface area equals the perimeter times the height. Plus two times what's our formula for finding the area of the base on this length times width, right? So what is the perimeter of the base? 8 times four is 32. Times the height, the length that connects them is ten plus two times length times width 8 times 8. 32 times ten is going to give us 320 two times 8 times 8 is going to give us a 128. Add them together and we get 448 centimeters squared. All right, there you have it. I know today's video is a little bit long, lots of vocabulary. It's going to get easier as we go on, all right? We're going to find lots of different shapes and surface area and volume and lateral area and all these fun stuff. Best of luck on the mastery check. Welcome back, mister Sullivan.