friends Romans geometry students lend me your ears welcome to lesson 12.2 we're talking about surface area of prisms and cylinders today here's some new vocabulary for you make sure you understand those terms by the end of this lesson go back in your book and review those terms as necessary if you don't know what those things are by the time we're done let's check out some definitions here to start this lesson so a 12 points you surface areas of prisms and cylinders here I'm starting off with some examples of prisms I'm not going to have you try to draw all of these here I'm going to refer to these as I describe what things are definitely a lesson we want to see lots of pictures with in terms of helping us better understand what things are so a prism what is a prism first of all well to polyhedra a number of polyhedron is made up of all polygons there's no curved parts to them so it's a polyhedron no curve parts no curved faces are allowed with two congruent faces and call those spaces they'll be the basis that lie in parallel planes so on these pictures just to make that make a little more sense hopefully I'm talking about this would be a base this triangle part right here and what's parallel to this triangle part will this triangle part these are also congruent to one another so here's a base and here's a base and the rectangular prism or the cube you could actually use this one and this one is the base or this one and this one or the front in the back is the basis any one of those will work because they're all congruent to one another and they're all parallel to one another so I could say for instance this one is going to do it to this one right here but I didn't have to choose those two usually they're on the top in the bottom but sometimes they'll rotate the picture around you have to know there's a parallel parts here the basis would be this Pentagon here is parallel and congruent to this Pentagon here parallel plane so here's your being here you should be here here you've got the hexagon the key of you could use right and left from back top and bottom any of those combinations would work but here this doesn't happen be it looks like it's not a regular hexagon hexagonal prism this one might be a regular pentagon offers and this looks like this might be the same all around here but not so much with this one I don't think this could be an equilateral triangular prism as well but from there I do want to show you something in the book here you up a little ways from there and we are and so I'm not going to have you draw these pictures in your notes unless you want to if you'd like to include them totally fine let's just talk briefly about what we see here so we've got these parallel planes you have a base here in the face here here you have a base here in a base here you can have a right rectangular prism so what happens to be rectangular because all these faces are rectangles the bases are also rectangles that's what makes it a rectangular prism because the bases are rectangles this is triangular prism because the bases are triangles here the height is straight up and down they can have slant heights as well if they are oblique oblique just meaning kind of tilted in a geometric sense so right rectangular prism goes straight up and down it's perpendicular here to the basis so these edges they go straight up from one base the next perpendicular in other words to the bases the lateral faces are the faces that are not the bases going kind of around the shape here the lateral faces would be these blue parts going around the shape here the lateral edges are the edges that are not part of the bases and so just know those terms understand those terms go back and review if necessary there but that's what we got going on there let's define a couple more things surface area you definitely want to know what that is if you're one of my students we very likely have already had an introduction to surface area in volume to kind of overview this chapter chapter 12 but this is going to go a little more in depth now so you might see for service area might CS a my CS period a period or you might do see the letter s capital s so know those things in this chapter will stand for the surface area and what is the surface area what's the sum of the areas that would make sense it's something to dealing with area but it's a sum of the areas of all of the faces so on here I'd be finding the area of this triangle or this rectangle right here this one right here that rectangle this rectangle right here and these two bases these triangular basis and i would add all those faces areas together that would give me the surface area of that total thing same idea holds with all other prisms with all cylinders as well the area of the surface is the surface area they of total faces all put together and the lateral area which we might refer to at times as la or L period a period is going to be the sum of all the lateral faces and by that i mean that no bases are included no bases are included so here with this triangular prism back to what we were talking about here the lateral area would be everything but this one in this one it be this rectangle out in front the one count of two the back right and the one to the back left you would add those three rectangles together but not this one not this one not the areas of those now let's check out theorem 12 points you from this point so here's one of the big things to know I don't want another example in one quite yeah I'm going to include this down here so surface area or sa we can refer to that as of a this has to be a right prism so surface area have a right prism this is theorem 12.2 by the way so we are talking about that here this is a really big important concepts I'm going to box that in this is one of the biggest two things for this lesson is knowing that this is a case so you have the surface area s is going to equal to x capital B capital B standing for we talked about this in our intro notes so I'm not going to write it again here but capital B stand for the area of the base you may want to make a note of that and pee standing form a perimeter and H staying for height you know what i think i am going to make a little note of each one of these things here so in a different color this is the area of the base let's write that down area of the base that's what capital B stands for it may have been a little while since we've done these introduction notes here this stands for the perimeter of the base and then right here the H stands for the height and then this ph part this whole thing together ph is going to equal the lateral area because it's the it's taking the bases and subtracting ones get rid of those this is just a lateral area part now why does this work why let's just think about why this works before we move on so if I were to find the area of the two bases coming back to this one I'd be finding this part right here and this part right here okay so that gives me of the five phases i have here that gets me that part in that part i could find the area of each one of these rectangles separately i could find this as two in length times width times height however you wanted to think of that the length here the base times the height of that rectangle you can do the base of this one times the height of that rectangle you could do all this separately but doing perimeter which would be all the way around doing that times the height does all three of them all at once if you did the perimeter of a pentagon it does all five of them all at once all five of these rectangles here he is so on if you did perimeter of a hexagon it would do all six of the rectangles that would be made and do all those all at once then be taking all that lateral area faces the lateral faces areas taking those with pH put them all together and adding them together with two times the area of the bases there's two bases so we do two times B instead of just a plain old be so pH is a lateral area H is a height p is primitive the base B this is the area of the base moving on from there let's check out an example so on these ones we're going to find the surface area will use abbreviation for that so I'm going to have you draw a picture here start by drawing a rectangle we're going to draw a rectangular prism on this one let's start just by drawing a rectangle but a couple different ways you can approach this this is the way I tend to do these myself I've just gotten used to do it like this i go back at the same degree measure same measure like 45 degrees ish or so is what i've done right there and i'm just going to go back connect those you can also do this you don't have to draw pictures here but you could also draw two congruent rectangles one behind the other as long as you're using pencil because we're going to have to erase some of these marks and then you could connect the points here here and here and here that will get you your rectangular prism I'm just going to go down from here draw a point right there and I'm going to draw the dashed lines in right away so I just went straight down at a 45-degree angle there made a little dot and now I connect those and that does the same thing for me as I did in this way this you just have to make some hidden lines here to make that pay legit picture and so from here let's draw in some measurements let's say this is 16 centimeters this one's four centimeters and the height is going to be nine centimeters and so that's the formula we're using well let's go back here if we forgot surface area is two times the area of the bases plus the perimeter times the height so first of all let's determine what our bases are usually the different-colored let's call the top and the bottom of the bases these are parallel parts they lie in parallel planes this rectangle in this rectangle and they also congruent to 1 each of one another so that fulfills the requirements for the basis for the prism so we're going to use this and this as our basis we're using s equals to be plus P times H so b is going to be looks like a rectangle would just be the little base times a little height of that rectangle there it's going to be 16 times for would be the area of each base now the perimeter we would take 16 plus 4 plus another 16 back here plus another four you can think of that as 16 x 2 plus 4 times 2 but i'm going to write the whole thing out and let's do that for now just so that we know where that comes from this is the perimeter of the base 16 plus 4 plus 16 plus 4 and we multiply that times the height the height is just part that's connecting the two bases so here so that height is nine right there so here I have two x 64 this becomes 40 right here and this becomes a 9 i can multiply this out I'll get 128 times it's going to be 320 right there and so doing all that I'm sorry i'm going to say 360 right there not not see that was an eight for some reason I don't know why so 128 plus 360 would give me 488 now this is the surface area area is always in units and we have centimeters here as our units of centimeters in this case centimeters times centimeters we're multiplying centimeters times centimeters that's centimeters squared so we have four hundred eighty eight centimeters squared for the first part of example one here this is still a prism here just as it's going to be over here we're going to draw a triangular prism for this one though and we'll make it an equilateral triangular prism so I want you to draw an equilateral triangle I'm looking at it a little bit of an angle so I'm kind of tilting it down as if you were looking at it on an angle there and then come straight down here this is looking just like I had right here I'm trying to just recreate this triangular prism basically and so connect that right there I know that this is going to be commute to this so you can where you could have drawing a triangle here the triangle here that's canoe it connected them and nidhi race I'm just going to go straight down though I'm going to I know my vertex will be right about there if I go straight down from there I've heard sex there connect those with dashed lines and that gives me what I'm looking for and so let's put this on the picture as well that this and this and this are all congruent that will let you know it's an equilateral triangle and I'm going to tell you that the side length is 6 inches right here and that this is 10 inches right there and so with our different color let's label this the capital beat this is my base this is my parallel and congruent base right there and so I can use the formula or the surface area of prisms based on that so I have s equals to be plus ph let's plug in what we know I know that the base well that's a little tricky here but I know it's the side length add this 4004 equilateral triangles it was side leg squared times radical 3 over 4 and so I'm plugging in 6 from my side length I have 6 squared times radical 3 over 4 so I replace be with the area formula that works for it here the area formula for rectangle is just base times height little be4 base their little based on little height would be 16 x 4 but this area a little more complex we have this shortcut formula so we can use that now what's the perimeter of one of these bases well that's just 6 plus 6 plus 6 so let's write that down 6 plus 6 plus 6 yes I know that's the same as 6 times 3 but just so we get the idea that printer we're adding the sides for summing up we are adding the sides together times the height here is 10 let's make a little notes of I think this one is pretty straightforward but where this came from right here it's like making a little note of that up here this is the shortcut formula s squared radical 3 over 4 for equilateral triangles that's worth that value came right there find me so let's simplify now we've got two times 36 radical 3 over 4 plus this will be 18 x 10 this I could reduce down to this would be a 9 and a 1 so i could say that's the same thing as 2 times 9 radical 3 plus 18 times 10 now I can do my multiplication here i'm going to have 18 radical 3 plus 180 this would be an exact answer you couldn't combine a radical part with a non radical part so we could say give you an exact answer this would be the final answer this in inches squared if you approximate this though put this into a calculator 18 times rad 3 that's about 18 times 1.7 and that ends up being about 31.2 31.2 plus this ends up being 211 point to this is an inches again this is an area so inches squared so exact answer is the top one approximate answer to one decimal place is what I decided to go with here is the bottom just make sure you check what they asked for they asked for two decimal places use too fast or three use three last for ten use 10 to ask for 1.5 decimal places you can laugh at them that's ridiculous but yeah it should be a whole number okay so a cylinder now let's get to that we talked about prisms now we're going to talk about cylinders to finish up the lesson here so that's a solid now it's not a polyhedron because all of these edges or all of these faces have curved edges some of them have curved edges at least so you've got this is a soup can a a thing that comes to mind for me when I think of cylinders that I see in everyday life a water bottle this general part except for this top part is generally a cylinder here so you got these circles right here and here that are getting connected here and here you've got a circle here and here connected straight down like that so it's a solid with congruent circular bases this time basis must be circles if you have a cylinder we're going to circular basis that lie in parallel plates and this is the other big guy from this lesson so I'm going to box this in it's a surface area of a we're just going to deal with right cylinder so straight up and down perpendicular bases are perpendicular to one another so surface day I have a right cylinder this is theorem 12.3 and we can use the same formula actually so we can start with this we can think of it like this its to be plus pH because you still have bases just like this in this we're Pentagon basis the same idea holds we're going to have the area of the base two bases here times the lateral area part but we have a different way to write that or an equivalent way to write that that makes more sense or is the way we need to think about it when we consider cylinders instead so a two V Plus pH which becomes so we can say this or equivalent way this is the really the way we need to think of it my bases are circles so think of it like this and I do want you to draw a picture for this one off to the side so let's draw a picture you're going to try to make our best cylinder right here as we can so I'm going to go straight down there if you using a pencil you can draw another congruent circle as best you can but these are ellipses the way I'm drawing them because they're being looked at at kind of on an angle here so let's connect those straight down and if you use pencil you could erase this like that if you want another way to draw this a way that also would work I sometimes dry mind like this where I'll draw circles and I'll draw a half circle there half ellipse and then I'll just connect that part there so that's a different way to draw it that's typically how i draw them on but either way is fine these aren't perfect I'm not looking for perfection right now with the notes I'm looking for that you got the idea down so this or equivalently i should say they're so equivalently well what is the base the base is a circle they're a different color let's put that ya gotta be here this is a be down here and since the base is a circle we know the area of the circle is PI R squared so that's where this next part is going to come from two PI R squared two basins two circles two PI R squared plus the perimeter of the circle here in the perimeter would equal the circumference circumference with a capital C that's what that stands for is 2 pi that's supposed to be an R that there we go that's better so perimeter right there is the same as circumference which is 2 PI R the base area is PI R squared and so the P i can replace with 2 PI R and we have that still times H this is really the way to think about it so this is what I want you to know rather than this but you can think back to this it's the same idea as with prisms it's still to be plus P times H so the idea remains the same it's just that we use PI and are because we're dealing with circles here now and so what is our come from where does H come from let's finish this up here in the picture this would be my radius my little R and the heights would be from here so here straight up and down we're saying is the height the H right so from there let's check out a couple examples so go back rewind refresh your memory I'll review that as need be make sure you understand the concept if you remember this concept or if you know that the concept is just to be plus pH you can derive the formula from that it's a lot easier to remember the formula based on that but example 2 let's draw a cylinder here so I'm going to go with the way I normally draw cylinders i'll usually just draw an ellipse represent my circle i'll go down like that and that and then i will make half of a circle or have it lips there and i'll make a half dashed and again you can draw you have to drop both you but you can draw two circles connect them and then erase the lower parts here to make that part hidden either way will work I don't need both ways though so i will get rid of that for now okay so i've got here let's say i give this information to you that the height is 8 meters and the radius is 4 meters and i'm going to tell you that's perpendicular there so therefore we know what's a right cylinder so we can go back to the formula i'm not going to use this although this is the idea i'm going to use this instead because that's how I really need to plug things in so I have s equals to PI R squared plus 2 pi r h from there plug in your values you've got your radius of 4 and you've got your height of 8 so it's pretty straightforward problem once you know what the equation is we need to get used to giving both exact and approximate answers so exact answers will be left in terms of pi here you have sixteen is four squared and then 16 times to be 32 so you have 32 pie and then over here you're going to have 2 times 4 would be eight and eight times eight would be 64 so you have 60 for pie right there now adding these are defining these together you have 96 pie so this is an exact answer if you were told to give an exact answer you would be done you would say that your answer is 96 pie meters squared however we should know how to get this as an approximate answer as well your calculator has a PI button that's definitely will we should be using rather than three point one four three point four point four is rounded off a bit too much in my opinion but we're going to use this now instead so my calculator has tons of decimal places now give me a really good approximate answer of 301 will go to two decimal places for now 300 1.59 so three hun point five nights so use that pie button on your calculator rather than just typing in 3.14 gives you a better answer okay so these would be exact values approximate values let's do one more example here before we move on to our final idea I had sub cylinders where I final India but there's actually one more thing I want to talk about briefly after that called next so on this one let's draw a kind of a wider fatter you might say cylinder so like this and we're going to say that my height is in this case my height is unknown oh man I don't know the height so a question back there I do know that all the way across is 19 inches and I'm going to tell you that it's given so here it's given the diameter is 19 inches and we also know that the surface area I'm going to tell you is 925 point 2 inches cubed we're going to try to find the missing value we're going to try to find how tall or how high is that so find the height or in other words find H so how can we do this how can we solve this use by a solution so I know about the diameter is 19 inches so since D equals 19 how can I find the radius which is what i need for my formula excuse me i know the radius is going to be 19 / 2 i'm just going to cut the diameter in half that would leave me with 9.5 let's put that in our different color right here that's something that wasn't originally in the picture but that I can determine based on the information given in the picture so 9.5 inches that's important not a final answer but it will allow me to get to my final answer so I can say at that point therefore I know that the surface area i'm going to use the formula with just the general way first so s equals two pi r squared plus 2 pi rh your s value is right here 925 point 2 i'll use parentheses as i plug stuff in here so you know that it was coming from up here so this value gets plugged in right here your radius 9.5 that's going to go right there and also right there it shows up twice and the height well we don't know the height so we're still calling that I don't know what that is but i can simplify this here and i can solve for h KH by itself with what i have left so I've got nine and twenty five point two nine point five squared would be ninety point two five and then 9.5 x 2 that gets you back to your diameter would be 19 pi so you have 19 pi h pi times B is the same as 2 times pi times R 2 pi times radius is the same as pi times the diameter so we have 925 point 2 equals this times to a be on an e point 5 pi we have that they're still plus 19 sorry I left out something important there the pie so 19 pi times H so from this point I would want to subtract trying to get h by itself remember pi is just the value it's a constant value it's 3.14159265 and so on and so forth so that times that three point one four ish can be subtracted from both sides so again I'm trying to leave this as an exact value as long as i can so i have 925 point 2 minus 180 point five pi is going to equal 19 pi times the height now to get the height by itself i can simply subtract not subtract sorry / then value in front of that 19 pie so if i divide both sides by 19 pi divided this by 19 pie bat will cancel here and here on the right side i'd just be left with h and then if we type this into a calculator let's get that calculator out and let's do that right now so i'm going to have 925 way to minus 100 point five my calculator goes order of operations so it's going to take this x pi first it's smart make sure your calculator is smart like that too before you would move on so that's what the top value is so that's what the numerator is 350 8.14 i need to divide this by not just 19 but 19 times pi so i need to put 19 and pie in parenthesis otherwise it's going to divide by x 19 and then multiply by PI instead of multiplying by x 19 first and then dividing the value by 19 5 so let's put 19 times pi in parentheses here okay so that will give me fifty nine point six nine now I'm going to take the value I had divided it by this I hit in equals okay so very very close to six who round that to two decimal places it would be 6.00 so we've got pretty much six inches right on the money there so approximately six inches for the height so we have successfully solved for H from there these kinds are a little tougher to little trickier you have to get h by itself if you round it off here but you mount it off and used enough decimal places you would get this value and better to leave it in exact terms though as long as possible till the very end there's one more thing we're going to talk about some nets here so I actually want to show you a video a very short video to go along with this to show you what nets are and how they come together we may have already created nets at this point you had the regular polyhedra where we had for less than 12.1 in lesson 12.1 you made these shapes we looked at these shapes you may have if you're in my class we may have made these already at this point sowwy we have the nets they were the unfolded parts the nets were the unfolded parts and then when we follow them up we made these regular polyhedra by doing that but if you go to some cool websites that i'll show you the net speed folded up in eating the solids in action so here I've got that and I'm just going to kind of skim through this I'm not going to play the whole video here but I'll keep talking in the background so you've got this 3d shape that's going to be made with this two-dimensional net so if you fold along the edges here it shows you all very nice it makes a QB fold along the lines that were there you're a little tough to see but we've got something like that so as they continue here it looks like they're getting a little will test your what shape do you think you could make based on this net so if you fold that up you fold that up before these up looks like that person got it wrong they clicked rectangular prism it's not that because you fold up the triangles these are going to be the parallel parts this triangle and this triangle right here notice how that and that would be parallel kind of got this tint looking shape but you have these rectangles as other parts well let's see I think we'll just go through this as well and this is a good little set so what do you think is this a square base pyramid cube triangle yes you have a square for the base here you fold up the sides boom there's your square pyramid right there build them like John Madden boom all right you know John Madden plane at men NFL like stuff you got that they're okay woods what's the basis here this is going to be parallel to this part when you fold it up looks like they got the wrong thing they're not a hexagonal prism yes octagon prison ok so we fold up all the sides it would look like look like that so there's a net in action being folded up to make a different shape they got how many faces does this have I don't think that's what we're too concerned about right now but you count up when I we count all that we'd have 10 yes ok so I'm going to stop that there how many edges you could count off the edges to that was the previous lesson and so that's what I'm going to show you would net so if you want to draw some examples from Nets you can if you want to look also at at page 7 29 you can see some more example of nets in your book i'll briefly show you what that looks like here so here they have a a net of a regular hexagonal prism they fold this up you get something that looks like that or if you look at that video you can just google search yourself where you're folding nets how did i type that in I think I typed it in as video showing Nets folding if you want to check that out this is what came up for me towards the top of that search some pretty cool stuff there to help you visualize what's going on so that's it for this lesson hope you found that helpful there's some trickier examples with ones like this remember just keep it exact as long as you can to solve for the missing parts and leave things in terms of pi as long as possible and then approximate from their use the PI button on your calculator on prisms the same idea holds for prisms it's to be plus ph four cylinders same thing it's to me plus ph what the b is PI R squared it's the same thing as a area of a circle p is a circumference pi r two PI R okay have a good day everybody and I will catch you guys later