11 2 Surface and Lateral Area of Prism and Cylinder

today we are starting new chapter chapter 11. we're actually skipping 11.1 so we're going to start with 11.2 surface and lateral area and we're going to start with two three-dimensional figures the first is prism and the second is cylinder so our objective by the end of the day is that by the end of the day you will be able to find both lateral and surface area of prisms and cylinders however another objective which isn't stated here is that as we are now in chapter 11 we are quickly coming to the end of our time together in geometry and have up until now always provided with you with template notes that is notes that have blank spaces where you need to just fill them in so you know exactly what notes to take that however is not the way things are going to work starting next year for you in Algebra 2 and certainly as you go forward in college you won't be given these types of template notes or generally will not so you'll have to learn how to take notes on your own and so now now that we are ending our geometry time together am not going to give you template notes to help you get ready for next year but hopefully it will be pretty obvious where you need to take notes and so if you need to you can always just pause this video and take your own notes for chapter 11. all right so let's get started here so we're going to start with 3d figure called the prism because it's pretty much the easiest figure to start with and there are lots of 3D shapes so we'll start with the easiest the prism okay the definition of prism is 3d figure with two bases in parallel planes and all the other faces are rectangles all right so there's an example these two triangles the one in pink and the one at the bottom of the shape are the bases the things are any any of the polygons that make up the prism are called faces so that face in yellow that rectangle in yellow is face and the base is face if face is not base it's called lateral face so every polygon is either one of the two parallel bases or it is lateral face and then the lateral Edge which is sometimes called the altitude it's the edge that makes that goes along the bay that goes from one base to another along the side of lateral face and in right prism like the one we're looking at here it's also the altitude the height of the shape and there are lots of different types of prisms and we name them based on would name them after the base so the example here would be triangular prism because the shape that the make the bases are triangles you can have square prism rectangular prism pentagonal hexagonal octagonal whatever any of the shapes we've learned in the past any polygon you can make it into prism all right and perhaps the most famous prism is one like this this is picture of Pink Floyd's Dark Side of the Moon their album but it is it is optical fact that if you take white light and shine it into prism the prism will actually break up that white light into its seven component parts that make up the rainbow of visible light and one of the things prism does is that all right so again let's go through the definition of prism as 3d figure with two congruent and parallel bases this by again might be time where you want to take some notes the bases are the polygons that are congruent and parallel to each other lateral face are the faces that are not base one of the polygons that makes up the prism that is not faith that is not base the altitude or height is the perpendicular segment that joins the planes of the bases or it's also the length of the altitude all right moving on Wright versus oblique prisms that so far we've only talked about right prisms on the previous slides we've talked about right prisms that is prisms where the base and the lateral face make right angle and that'll be the type of prism we do almost all the problems on however not all prisms are right prism May lean little bit and if so it is called an oblique prism so again right prism is prism where all the lateral faces are rectangles and lateral Edge is an altitude all lateral sides are perpendicular to the base as it makes it right prism the fact that they're perpendicular to the base and oblique prism sum or all of its lateral faces are not rectangles they could be polygons they could be parallelograms and the base and lateral sides may not be perpendicular so let me show you graphically what's going on here so there is right prism again how it looks like box and release this one because it is it is rectangular right prism has 90 degree angle there is similar prism that is an oblique prism that is not 90 degree angle and so if want to find the height of this prism like there or there the height of the prism is the just like when 2d in 2d figure it must be perpendicular to the base so the height is the distance from one base to another that is perpendicular to both bases so it may actually if it's oblique prism like this the altitude may be outside of the prism as it's drawn both times here all right let's talk about surface area surface area is just the total definition excuse me the total area of all the surfaces of 3D object if you add up all the the faces of prism you get its surface area so for example right here is rectangular prism if take the area of the top and add that to the area of the front as well as add in the area to the left as well as add in the area at the bottom and the back and finally the right if add up the area of all six of the faces of this rectangular prism will get its surface area you just have to add up all the faces here's another example of what's called pyramid which we'll get into in little bit here but this is this pyramid also has where we're adding up you can see the bases of this pair excuse me the faces of this pyramid are all being added together to get the surface area the lateral area by definition is the total area of the lateral faces of the 3D object so you can think of it as it's you can either add up all the lateral faces or you could take the entire surface area and then just subtract the area of the bases all right so if have that shape it has these five faces the front is part of the lateral area the right is part of the lateral area the back is part of the lateral area the left is and the base is not it would not be part of lateral area but of course the base would be part of the surface area right to do surface areas one way that one can do them or the one way to suggests is to be able to do nets that is you need to be able to visualize all of the sides so imagine if had this shape here this is rectangular prism and imagine if it were sort of like wire frame and then with the pieces of plastic put around it as it's done right here could unwrap that plastic so here you can see I'm sort of pulled the blue side down little bit I'm starting to pull off the the side faces I'm continuing that doing it more and more and now finally you can see here that this is what's called net and you can see why this would be much easier to figure out exactly what the surface area is by making it two-dimensional by taking this three-dimensional shape and taking each of its each of its component polygons and putting them together in plane so that can see them together you can see how that would be useful to try to visualize the assign and use it to try to find the surface area or lateral area okay so that's called the net of solid and again it looks something like that so you can see exactly how to calculate the net of solid or something that the area of solid by each of its different pieces with its length its height and its width all right want to show you an example of so this is net of solid right here look at it right here and see if you can in your head put this together and figure out what kind of solid it is fold it here here here here and here and this is rectangular prism all right let's do another one do it the other way around let's start with rectangular prism and turn it into net so if you take this one and it can unfold it unfold it and finally the last two pieces come down and you can see how that was that net there here's another example of net that's unfolded different way and again another net there all right finally here's little quick here's here's quick quiz for you to see if you can figure it out which of these three shapes or is going to be the net of the shape over there on the left you have about 10 more seconds to figure it out in your head see if you can sort of fold up the shapes and see which one of them is going to make the actual shape there and the answer would be would make the same shape all right so again you need to be able to visualize the sides of net and once you do as it is as this example here shows then it's relatively easy to figure out the area of all the different pieces so if asked you to find the surface area of this of this one here you would just have to find the area of each of the six different sides by taking the formula of width times height and multiply them out and you could add up all the individual pieces to come up with your surface area all right so try these example problems example problem one and you should post pause and try them right here and once you have done them or get stuck on them then come back and continue with the video and will show you how to do them now all right so number one is asking us to use net and so if were to take this shape and unfold it if you will it would look something like this where there's that four feet that would go here and here and here and here the other four feet would go here and here and there and then finally the 10 feet would go here and there so now can use this net to calculate the area of each of the individual sides of the prism so the one in the upper left that 4 times 4 that would be 16 square feet four times four feet times four feet is 16 square feet that little square is also 16 square feet four times four and now have 10 by 4 10 by 4 lateral face 10 times 4 would be 40 square feet another 40 square feet another one and finally last one and so then just need to add all these up the two sixteens and the 440s when add all of those up together get final answer of 192 square feet all right when look at number it's not asking me to use net just to use the formula so when look at this see have two bases that are right triangles can see that they're right triangle because that one on the bottom there has right angle in it and of course the formula so the right triangles with base of three inches and height of three inches and of course the area of right triangle is the or any triangle is base times height times two and can easily figure out the base and height use because it's right triangle it's three times three divided by two so the area of one of those two right triangles would be 4.5 square inches okay there are also two lateral faces that have base of three and height of 10. those are the ones that are partially obscured by our view on the left and the right and of course the area for rectangle is just base times height so that would be 3 times 10 so the area of each of those two lateral faces on the left and the right would be 30. and then finally I've got one more rectangle which has height of 10 and base of this red line right here excuse me this red line right here and don't know what that is need to calculate it but luckily have this makes up this triangle here and it's right triangle and so I've got three sides and and of course we know our Pythagorean theorem that squared plus squared equals squared so can use that to find my side so it's three three squared plus three squared is squared so that would be 9 plus 9 is squared or 18 is squared so must be the root of 18 which if put into calculator is equal to about 4.24 could also have done this using special right triangles because it's an isosceles right triangle know that the ratio of the sides is root 2. in this case three three and three root 2 and of course three root two is four point two four so now that have the base as well as the height can use that know the base is 4.24 as we just figured out and the height is 10 so can multiply that out and get to the area is 42.4 all right now that know the area of each of the individual sides just need to add them up so I've got two triangles that are 4.5 each so when multiply those out to 4.5 is 9. the two lateral faces that are the two lateral faces that are three by ten are both 30 so two of those would be 60 and I've got the one final one that's little bigger that's the lateral face is little bigger it's 42.4 when add up 9 60 and 42.4 get our final answer of 111.4 square inches all right the next problem again once again suggest you pause try these problems on your own and then you can push play to see how you did all right it's asking us to classify the prism well the bases of that prism are pentagon so the base has five sides therefore we would call it pentagonal prism the bases are regular polygons find the area of the prism I'll find the lateral area of the prism excuse me find lateral area Okay the lateral area is made up of five rectangles each one has base of five and height of 11. so that would be 55 for each and we have five of them and so 5 times 55 would be 275 square centimeters okay and then part asks if the area of each base is 43 square centimeters find the surface area of the prism so the surface area is simply the lateral area plus the bases so if have pentagon that's 43 square centimeters then two of them for the two bases would be 2 times 43 or 86. and if add that to my 5 lateral faces which we just figured out earlier was 275 add those up to get the surface area and the surface area would be 361 square centimeters now let's talk about different 3d figure called the cylinder the cylinder is 3d figure with two parallel circular bases connected by curved surface we were just we just finished with prism and you can think of cylinder kind of like prism they're similar to prisms because they have two parallel bases that are connected the difference that in the cylinder that in cylinder the base is circle not polygon so it looks something like this and in I've defined it and talking about it as circle and cylinder may have oval bases but we're only going to look at circular ones so we're just going to talk about circular cylinders whenever say cylinder mean circular base but to be technical about it any cylinder may have an oval base as opposed to just circular one all right and so those are the bases that's the radius of my circle and that is the height of my cylinder okay and cylinders are everywhere right when you have cans cans or cylinders oil drums or cylinders there's Drums of stuff toilet paper roll or paper towel roll right all these are cylinders and my favorite cylinder is of course if you're Muppets fan Beaker is cylinder with little hair on top all right so there is cylinder and just like we had write an oblique polygons excuse me prisms we can have right and oblique cylinders on the previous slide we talked about right cylinders that is they go straight up and down there have right angles however not all cylinders are right prism May lean if so is it called an oblique cylinder so right cylinder is cylinder where the segments joining the centers of the bases is an altitude and they're perpendicular to the base and oblique cylinders where that's not the case cylinder or the segment joining the centers is of the bases is not an altitude it is not perpendicular to the base so for example there are the examples there on the left have right cylinder and you can see that the radius and the height and then and the line makes that makes straight line between the two centers of the bases is at right the angle as opposed to the oblique cylinder that is not the case that it leans and just like just like the previous slide where we talked mean just like when worked for prisms where we talk about you need to be able to visualize Nets you need to be able the same thing with cylinders the net of every cylinder is exactly the same thing you've got the bases or both circles with radius of we know to find the area of those is just pi squared and then the remainder is piece is rectangle again if you take piece of paper and you roll it together so you take one side of the paper that it is that it touches the side that is parallel to it you have made yourself cylinder without top and the bottom and that piece of paper the height of the paper is how the height of the cylinder and the width of the paper is how long it is all the way around which of course because of the circle would be 2 pi so every cylinder has very similar net it's got two circles and then rectangle that is the definite that the dimensions the rectangle or the height times 2 pi all right now that you know that let's do these two example problems the first one problem three is asking us to find the lateral area and in the surface area of this cylinder so push pause and try this example problem on your own and then come back and see how you did all right problem number three is asking us to find the lateral area of the cylinder so let's draw little net of the cylinder like the one just took from the previous page and right there is going to be my lateral area right the lateral area is everything except the bases so to do that what we know is that the area is the base times the height so it's just rectangle and in this case that would be 2 pi times the height and so we know the radius is 5 meters and the height is 11 meters so that would be 2 pi times 5 meters times 11 meters put all that together and get 110 Pi square meters so that would be the lateral area of this cylinder and if wanted if had to round it you could just you know multiply by 3.14 for pi and get whatever you needed to get all right the next question is asking us to find the surface area of that cylinder well the surface area is just lateral area plus the bases so there's the base one and there's base two and so need to find those and so the area of circle is just pi squared so that would be Pi 5 meters squared so each of those bases is 25 Pi square meters since have two of them would multiply by 2 so that would be 50 Pi square meters and as just said the surface area equals lateral area plus the bases so that would be 110 Pi meters squared plus 50 Pi meters squared that would equal 160 Pi meters squared four square meters all right let's take look at problem number four suppose that cylinder has radius of units and at the height of the cylinder is also units the lateral the lateral area of the cylinder is 98 Pi square units given that information first find the value of and second find the surface area of the cylinder okay so push pause and try this problem on your own and then come back and see how you did once again here's the net of the cylinder and this is the lateral area and we know that the lateral area is just rectangle of Base times height 2 pi times and so that would be 2 pi times because we know that both the radius the radiuses are and the height is so that and we know that that's equal to 98 Pi square units so 2 pi squared must equal 98 Pi if divide both sides by pi I've got 2 squared equals 98 divide both sides by 2 and I've got squared equals 49. take the square root of both sides and end up that equals 7. all right now to find know that the surface area is just equal to lateral plus the bases and here are my bases right here and the area of circle of course is pi squared so that would be Pi seven squared that would be 49 Pi square units so both of them together both bases together would be 2 times 49 Pi or 98 Pi square units so if the surface area equals lateral area plus the bases and it told us that the lateral area was 98 and we just figured out that the surface area is 98 Pi if add up 98 Pi unit squared plus another 98 Pi unit squared get that the answer is 196 Pi square units