Surface Area Cones Cylinders and Prisms Lesson

in this chapter we will be looking at topics related to measurement in this lesson we will be learning to calculate the surface area of cone cylinder and prism we will begin by looking at the surface area of cone okay so now we're going to take look at at cones cylinders and rectangle rectangle prisms here we're going to find the the surface area in particularly here so surface area is the amount of material needed to cover the outside of any shape so it's that that two dimensional measure like it's the amount that you can kind of feel if you will it's the it's the part that you you can touch right and I'm sure most of you are fairly comfortable with that idea of area so mean another way of thinking about it here is the amount of paper needed to wrap present that's actually really good way of thinking about that so we're gonna start off with cone okay now cone consists of two parts what you've got is that lateral area here and so if you think about it like if we were to yeah normally what you would do here is you would take like circular bit right here cut it off so it's kind of like pacman here and when you glue this and this together it causes that conical shape to pop out here so this is the part that we're looking at here that lateral surface area the the part of the cone like if it was an ice-cream cone that's the part of the cone that you're grabbing right that's the lateral part okay and the part that area right there the formula for that is going to be PI now just to be clear and you can see it down here pi obviously is PI is the the radius of the circle that's formed at the base of the cone okay and is the slant height it's the it's the distance from the the vertex of the cone the peak of the cone down to that circle that circular edge should say okay that is our here and so it turns out that the surface area around the cone the part that you've grabbed is just PI RS then you got to consider the base of the cone that circle in the area of circle is PI squared so putting those together we get there's total surface area of cone is going to be PI RS the lateral area plus PI squared the bottom now you got to be aware that in lot of cases you aren't going to be given all of the information that you need immediately to plug information into this oftentimes you're gonna have to use the Pythagorean theorem now just to give you an illustration when mean let's say you've got cone here okay oftentimes what will happen here is you will know you will be given this distance right here for example let's say that this is three and you'll know the height of the cone let's say that that's four but in order to figure out the surface area really it's it's not the height of the cone that need it's this distance along the side of the cone this is our so now how do you figure that out well you would use the Pythagorean theorem 3 squared plus 4 squared is equal to squared and it shouldn't take you too long to figure out is equal to 5 and then have enough information to figure out the surface area of that that cone okay so just bear in mind that sometimes you're going to have to use the Pythagorean theorem to figure out piece of the information that you might not have now let's take look at some examples all right so our first question here says to determine the surface area of each cone to the nearest hundredth okay well this is really light here what you're not seeing here is that this is 13 meters and this right here is 5 meters ok now 13 understand to be the slant height here so the surface area at pi RS plus PI squared is going to be PI well the radius of circle at the bottom is 5 that's pretty straightforward and in this case we're giving us as 13 and then we've already established that we know what the radius is and so now this is really just matter of going to the calculator and plugging that all in okay this will plug it in exactly as we see at PI times 5 times 13 plus PI times 5 squared press ENTER and we'll do this to the nearest tenth here we get two hundred eighty two point seven 282 ulcerated whoops see got distracted looked there it says to the nearest hundredth so was gonna do that wrong so it's gonna be two hundred eighty two point seven four this will be two hundred and eighty two point seven four meters squared oops to the nearest hundredth thanks there it is it's as straightforward as that when you've been given the slant height now take look at this one this case right here we know and again it might not be hard percent clear that that radius at the bottom is 12 and the height here's make that little bigger the height is 20 okay so now in order to figure out the surface area need the slant height so we know that 12 squared plus 20 squared will equal squared okay well 12 squared is 144 20 squared is gonna be 400 okay that's gonna be 544 is equal to squared now I'm gonna end up doing this to the nearest hundredth so I'm just gonna write this as the the square root of 544 okay so that's that is my my slant height here so now the area will be PI RS plus PI squared so pi the radius we saw was 12 and then the slant height was the square root of 544 plus 5 times 12 squid now I've got all the information just got to plug this into my calculator whoops so this'll be PI times 12 whoops times the square root of 544 plus PI times 12 squared okay and we're gonna get 1331 0.68 1331 point six eight units squared we never actually established what the units were over the diagram so that's about as good as that's gonna get there's your surface area okay so this question says here that some paper cups are shaped like cones how much paper to the nearest square centimeter is needed to make the cup okay again if you can't see it here this this telling us that the radius here is 8 centimeters and that the depth here is 12 okay now this is an interesting question because in order to make the the cone actually don't need that circle at the top here and so this is one of the things that you're gonna be required to to do here this is why surface area tends to be more complicated concept than volume okay because with surface area you've typically you've to poq ly got to piece together all of the different surfaces that make up solid okay now with volume it's just you just use the volume of that solid mean you might have different kind of shapes put together but you just do the the volume of those those pieces here here it's very much the case for your you're looking at the different surfaces and deciding whether or not you're including them in this case we are not including that circle that would normally be at the base of the cone because that would actually plug up the cup do however need to know what my slant height is and that's going to be 8 squared plus 12 squared that is going to equal the slant height squared and so 8 squared is going to be 64 plus 144 is going to equal squared so we're gonna get 208 is equal to squared and then I'll just take the square root of that and it'll be the positive square root know that you can get positive or negative square root here but it's got to be the positive square root and mean could simplify that the radical here but I'm gonna be plugging this in the calculator just second so might as well just leave it as the square root of 208 my area okay the surface area of the part that I'm gonna grab is simply going to be PI in this case pi times 8 times the square root of 208 don't need to add the circle because that that actually isn't part of this particular shape here so PI times what was it as eight times the square root 208 sorry don't need that okay and what does it say to the nearest square centimeter well to the near square centimeter that's 362 centimeters squared you probably need little bit more than that if you're going to tape it together and whatnot but that's it's basically what the question is asking for right so this question right here this is really good question that's really good assessment question because it's requiring that you go backwards with it - there's few things in here but but mostly it's about doing undoing the algebraic steps to solve for specific variable here so in this case we're given that the surface area bolla of the comb below is seventy five point four centimeter squared we want to determine the value of which in this case is the slant the slant height here of the cone to the nearest tenth now when you take look at the cone here notice that we're given the radius here but the radius is given in terms of millimeters we have convert that to centimeters and so 20 millimeters here that's that's two centimeters so now these straight units are consistent and we know that area is equal to PI plus PI squared now let's just plug in everything that we know well we know that the radius sorry the area here seventy five point four pi times two in this case the they're using the variable for that slant height plus pi times two squared now simply need to get here so I've got two terms on the right hand side really needed to get it down to just this one term with the in it so I'm going to subtract that term it doesn't have the exodus so minus pi times two squared sorry is equal to PI times two times and now I've got these two coefficients in front of the there okay there are two factors should say in terms of the coefficient I'm just gonna divide by that so seventy five point four minus pi times two squared over 5 times two and that should give me now it's just matter of going to my calculator plugging that in so I'm going to put parentheses around the numerator because the numerator consists of two parts so want to make sure that get everything in there so it's going to be seventy five point four minus PI squared or PI times two squared divided by and now because the denominator has two parts to it it's it's PI times two again need to put parentheses there if don't then the calculator will divide by PI and then multiply the answer by two so need to put parentheses there to tell it to divide by that whole thing press ENTER and we get this beautiful little value here okay is equal to ten point zero centimeters okay really really good question something you should have be prepared for on your assessments and now we'll introduce the formula for the surface area of cylinder now let's take look at the surface area of cylinder which really consists of two parts you got the two circles okay you got the circle on the top and the circle on the bottom and then there's this rectangular area that goes around the two circles now if that's not immediately obvious if you were to take the the cannula so you popped off the lid at the top and the bottom of can it just made cut right down the side straight down the side when you unfold that you would get rectangle actually actually think if you if you think of that if you were to take this thing and can kind of roll it up okay you would see that that that that rectangle there can become that circular shape or that cylindrical shape that we're looking for so the total surface area of cylinder then is gonna be the area of the two circles plus the lateral area bilaterally we mean side okay the sign area which is that that rectangle that we were just talking about here and so two PI squared just think about that PI squared is the area of circle so two PI squared is the two circles top one in the bottom one this one is little less obvious until you really break it down here is the height so if come over to this to this guy right here this is the height right there that rectangle now the other thing that we've got here is two PI naught where do we recognize two PI from well two PI is the circumference of circle so what they're saying here is that this top part here if it's to be cylinder this top part here has to wrap all the way around this distance here has to go all the way around that circle so that means that this straight distance here has to be equivalent to the circumference so that circle 2 PI and so there's why our service area formula looks the way it does and we also take look at some examples now all right now this question says that the base circumference of cylinder is 220 centimeters and it's height is 63 centimeters find its total surface area okay well here we go here's our piece right here now ideally what would have loved is to have had the radius okay because then could have just have plugged it into my formula but don't have that do however know the circumference now the reason why that's significant is because remember in my formula this was their circumference of the circle so don't even need to calculate that already know that that's 220 what don't know okay what don't know is what the radius is that's what needs to be looking for here so 2 PI is equal to the circumference and right now 2 pi is equal to 220 so now to get my radius will divide both sides by 2 pi now when divide 220 by 2 I'm gonna get at 110 dividing by pi is little more awkward here requires the calculator and I'm not going to do that right now but there's my radius so now can go to my surface area so my surface area is going to be 2 PI squared plus 2 pi RH so this will be 2 pi now squared is going to be 110 over PI squared and then 2 pi well 2 PI is 220 we already know that that's circumference of the circle so 220 times which in this case was 63 no when do this when put this together I'm gonna get 110 squared okay I'll just quickly go to my calculate 110 squared is 12100 so this will be to 2 pi 12100 over pi squared because the Pi in the denominator also got squared plus 220 times 63 which is 13,000 860 now one of those PI's is going to cancel and two times 12100 will be 24,200 over pi Plus 13,000 860 now without evaluating the pyre plugging in number for pi is arguably about as simple as that's going to get so what we'll do here is we will actually go to my calculator to four thousand two hundred divided by pi plus thirteen thousand eight hundred sixty and I'm getting surface area that is twenty one thousand five hundred sixty-three now rounded to the nearest tenth point one centimeters squared so it's approximately that value there okay there we go okay so this question we read that the height of we're gonna supposed to find the height of the cylinder the nearest tenth if the cylinder has radius of 6.5 centimeters and the surface area is 592 point one nine centimeters squared so this is case of us really just manipulating the equation here okay so this is the surface area of cylinder okay we already know that the surface area is 592 point one nine centimeters squared this will be two pi we know that the radius is 6.5 centimeters squared so this will be two times pi times six point five times are unknown so want to make this equation to get my unknown and this is something that that we really need to make sure that everybody is capable of doing here so there are two terms on the right-hand side of the equation one of them has the with it want to isolate that one so I've got to get rid of this one they are being added together so I'm going to subtract that so 592 point one nine minus two pi times six point five squared I'll evaluate this all later when use the calculator so 2pi six point five now on the right hand side I've only got one term with that one variable and that's what want to get by itself so am multiplying that variable by two by PI by six point five okay now remember when when you've got bunch of stuff multiplied together like this in row here they're all being multiplied together order doesn't matter can switch the order around over want so it's really just this whole bit here multiplied by so I'm going to divide everything by the two pi six point five so will get five hundred ninety two point one foot nine minus two pi times six point five squared and that will all be divided by two times pi times six point five that will give me my value for now I'm gonna go to my calculator now want you to see what I'm doing here because it is frequently the case that when the mistake gets made it's must make made because people don't really know how to use the calculator okay there is this bar right here which means division but if there's little bit more to it than just that that bar implies set of brackets around the numerator denominator and if you don't put that bar there you've done it wrong okay so if that put parentheses around the 592 point one nine minus two times pi times 6.5 squared and then got close that set of parenthesis for the numerator divided by and have to open them again for the numerator if you don't if you don't open that set of parenthesis for the denominator then you're dividing by two and then you're gonna multiply the numerator by pi times six point five lot of times people have no idea they don't even notice that the answer is is really really wrong okay so please make sure you put parentheses around this two times pi times think I'm off the screen now yeah six point five closed brackets and there we go so to the nearest tenth this is approximately eight point zero centimeters yeah seven point nine nine eight nine whatever so we round to the nearest tenth eight point zero and finally we'll look at how to find the surface area of prism now let's take look at what constitutes the surface area of rectangular prism so it consists of three parts here are you gonna have the top the bottom and then the sides now okay so we've got the length the width and the height here now the bottom is gonna be the area that's gonna be the length times the width it's this rectangle right there now that bear in mind that could be square this could be square square prism in which case actually sorry we're gonna talk about that just second here but so they'll be length times width there's gonna be two of those exact same sides there one at the bottom one at the top then we're gonna have the front face here which will be length times height and again there will be two of those like the front at the back and then there's gonna be square rectangle here that's that's height times width that's gonna be from the right hand side and then the left hand side okay and so what you get here is ya length times width there's two of those height times width there's two of those length times width there's two of those and you have to put those all together so and when you do you're going to get the surface in rectangular prism will be two times okay little bit of whoops here so two times length plus width plus two times length times height plus two times height times width is really what we wanted to say there okay whoops those that would be your formula for the surface area rectangular prism now if it's square prism or square prism or cube for that matter what you're gonna get here is that these these are going to be the same year now if it's if it's cube I'll actually show you this this shouldn't say square prism this should say really what they want here is is is cube surface area of cube if it's cube that each of those sides are exactly the same so this would be like like and then when you flip that over the right here and so what happens is the surface area of the cube would be six times squared now if it's square prism if it's square prism then what you've got here is you're going to have square on the bottom and then you're gonna have these rectangles popping out rectangle let's say rectangle rectangle rectangle and then you're gonna have another square someone right somewhere right there okay so this is maybe let's say so this is going to be the the length of each of these would be here and now the surface area okay would be 2x squared for square prism and then each one of those sides there would be four of them they'd be exactly the same will be times okay so surface area of cube is just gonna be six times the side squared let the surface say you have square prism would be 2x squared for the the top at the bottom of the squares and then for for the the lateral sides are those those rectangles that aren't squares that you have to consider all of those is that let's take look at couple of examples okay so this question says find the surface area the following and can tell by looking at this that this is rectangular prism because none of the dimensions are the same there's nothing repeated here so my service area is going to be two length times width plus two length times height plus two height times width okay so now really have to just decide which of these counts as length which one counts as width here and which one counts as height so maybe what would do is six feet is the length three would be the width two would be the height so this will be 2 times 6 times 3 plus 2 times 6 times 2 plus 2 times 2 times 3 and so what have we got here 2 times 6 is 12 times 3 is 36 2 times 6 is 12 plus 2 is 24 2004 times 3 is 12 and so 36 plus well what do we got here I've got these are all multiples of 12 I've got three of them here two of them here one of them here that's 6 that's gonna be 72 so 72 feet squared is the surface area that rectangular prism so in this question we breed that the surface area of cube is 486 centimeters squared we determine the side length okay well the surface area of butt of cube was 6x squared where was the length of the side so 486 is equal to 6x squared so now I'm just going to solve for that there so first of all going to divide by 6 I'm going to do that I'm going to get 81 is equal to squared and then got to take the square root of both sides so this is going to be the plus or minus square root of 81 which is going to be plus or minus 9 except that know that the negative doesn't make any sense in this context so it's just gonna be 9 centimeters is going to be the length of the side that's all really got to do now take look at the assignment you