Geometry 11 2 Surface Area of Prisms and Cylinders

hey class this is the first lesson of the surface area and volume unit and today is 11-2 in the textbook we're going to be doing the surface areas of prisms and cylinders it's just quick summary of this whole unit we're gonna be doing how to find the surface area and volumes for all but one of the shapes over on the right I'm sure you guys can probably guess which of those shapes we're not going to be able to calculate the surface area volume for but we are going to cover prisms cylinders was actually arm picture here cones and pyramids cubes which are actually just special type of prism and finally spheres which look like they might be kind of complicated but are actually really simple so the things you'll need this unit are the homework and mastery quiz list or you can just get everything off Google classroom and of course formula sheet for surface area and volume which we'll be passing out in class so in the the first thing you need to look at is what is prism so when we talk about prism we're basically talking about any shape that looks something like this where we have base on the side and they're connected by these what we call lateral faces so the lateral faces connect the bases the it for it to be prism the two bases need to be the same so notice here that we have like Pentagon on the top and the congruent Pentagon on the bottom in this base shape this is by the way in animation in your textbook you can actually move it around and stuff it's kind of cool we have hexagon on one side and we have congruent hexagon on the other side that's kind of important notice that prism can have bases of any shape so we have like pentagonal prism up here hexagonal prism down here triangular prism over here and of course our favorite is going to be the rectangular prism at least that's my favorite those are going to be easiest to deal with we can have right prism and these are going to be most of the ones that we deal with where the base is perpendicular to the height so this is kind of think what you would think of as like just box right where everything's right angle this is especially true in the case of rectangular prism that's like really easy to think of as cardboard box like something Amazon might send you over here that we have the oblique prism so what's happening here is that the base isn't perpendicular to the height so you notice that we've seen triangles like this where the height isn't one of the sides it's actually it's we sort of can see it best if we set it off to the side little bit so this height is still the height between the bases it's just that the sides are slanted so don't know this would be kind of bad box to ship something and think but but the height is actually outside the object or we can think of it as being inside but it's not gonna be one of these society sides because the bases are actually closer together than okay so the first thing we're gonna look at is using net to find the surface area of the prism we haven't talked about nets whole lot but want you guys to imagine that we kind of take this whole box and we just unfold so hopefully everyone can imagine like we've taken this whole box we just kind of broken it up into each of the different phases so what we're gonna do is calculate the surface area now of this box so one of the phases notice is five long and four wide we can find the area of that rectangle right we do five times four notice we're doing that here and then we do there's gonna be another one where it's still five tall but now it's three wide so we do five times three then if we go to the back of this thing it's five tall four wide five times four and if we look at this side over here that'll be five total three wide so five times three then we got to deal with the two bases right so four times three and again on top four times three so we're gonna wind up with 20 plus 15 plus 20 plus 15 plus 12 +12 that's 94 we're gonna find more efficient way to do this in sec don't worry here's that more efficient way so we're gonna talk about two kinds of area here first of all we're going to talk about lateral area now the lateral area want you guys to think about is the kind of side area of this thing so you could include this side this side on this back side over here that's kind of on the opposite side of the box and this side over here what it's not gonna include is the top and the bottom of the box so those would not be included in lateral area lateral means side so the top and bottom Rock can be included in the side area but the top and bottom are gonna be included in the surface area which is the sum of all the surfaces of this box so the lateral area of right prism is actually super easy to find it's just perimeter times height when we talk about perimeter here we are talking about the perimeter of base okay then the surface area of right prism is going to be the sum of the lateral area which we can find using this formula and the area of the two bases the one on the bottom and the one on the top let's see an example of this in ours so we'd like to find the surface area of the prism shown notice this is triangular prism that's gonna make things little more complicated but not hugely more complicated also want to point out this is right triangle and looking at this angle here that's also good an important first thing we need to do right off the bat is find the perimeter of the base notice we've got three plus four plus we don't know this side here but we can use the fact that this is right triangle to set up our squared plus squared equals squared and if we do that through we're gonna find three squared plus four squared is 25 and the square root of 25 is five so this last side is going to be five which means our perimeter is three plus four plus 5 gives us two now we've got to find lateral area remember lateral area equals perimeter of the base times the height so we got this 12 up here you're gonna multiply by the height that's gonna give us 12 times 6 is 72 okay so now we've got the lateral area which is the area of this side over here this side over here and this side in the front we've got to do next is find the area of the base so what we're gonna do that is simply use the formula for the area of triangle notice the bases are triangles so we're doing 3 times 4 divided by 2 3 times 4 times 1/2 and that gives us 6 notice 3 times 4 that's 12 divided by 2 that's 6 step 4 we need to find the surface area of the entire prism remember we found the lateral area but that only gives us the side areas doesn't include the top or the bottom so we've got to add now twice the area of the base so if we look at this we've got 72 is the lateral area 6 as the base area plug that in we've got 72 plus 2 times 6 and that should give us 84 centimeters squared now notice surface area is still an area calculation so we're still going to wind up with centimeters squared later in this unit we will see volumes and those are going to be centimeters cubed but we're not there yet we're just on the surface area surface area is going to be centimeters squared or anything squared okay let's talk about cylinders for second don't want you guys to think about cylinders as completely different from prisms because they're really not notice that the only difference is now the base is circle we still have base at either end the bases are still the same size they're congruent same size and shape in the height is the other dimension here again we have right cylinders meaning the height is perpendicular to the base and we have oblique cylinders where they're kind of slanted but same idea Potts we have base we could find the base area here using the area of the circle formula pi r-squared we have height so let's see how this all clicks together with the formulas the only difference here is that instead of perimeter we're gonna talk about circumference now want you guys to imagine okay you can think about this almost like if you took toilet paper roll right and you cut it right in straight line right down the toilet paper roll and then you kind of folded that out the shape that you get would actually be rectangle all right so they're kind of showing like mostly curling this all the way out and by the way this is another animation in the textbook which highly recommend you check out so if we rolled this out we'd get rectangle as the lateral area which would have length of 2 pi that would be the circumference of the circle 2 PI that's almost like the perimeter from the earlier formula and we're multiplying that by the height of the cylinder so notice the lateral area which was earlier perimeter times height is still gonna be perimeter times height it's just now we have better formula for perimeter so the circumference sorry the area of this whole blue thing is going to be 2 PI see if we can apply that they will apply it in second sorry so first of all we just want to review lateral area equals 2 pi RH right or PI notice that here is taking the place of two R's because the diameter is twice the radius the surface area is gonna be the lateral area plus again same as with the prism two bases or surface area is going to be equal to 2 pi RH that's this formula up here plus 2 PI squared which is 2 area of the circles so notice this formula not actually that complicated when you really get into it the only things that ever change in it are are the radius of the circle and the height of the cylinder so let's try to put this into practice ok the radius of the base of the cylinder is 4 inches so notice radius 4 and its height is 6 inches is 6 what's the surface area so we're finding surface area not lateral area of the cylinder in terms of pi so remember what that means we're gonna leave the PI in the answer notice we have PI's and all these answers so first of all we're using this formula for the surface area of cylinder surface area equals lateral area plus 2 bases that's 2 pi RH plus 2 PI squared now we plug in the numbers so it's going to be 2 PI 6 or sorry 4 times 6 plus 2 pi 4 squared now can already see this is gonna give us 48 over here and 32 over here so we're gonna get 48 PI plus 32 pi which think is going to give us 80 and indeed it does so the surface area of the cylinder is 80 PI inches squared which would be answer choice let's do another one this is little more applied think notice so you are using the cylindrical stencil roller below to paint patterns on your floor what area does the roller cover in one full ton so you can imagine this thing going like all the way around when you kind of turned it once how much area have you covered so they gave us now be careful here they didn't give us radius right they give us diameter so we're gonna try to use the the formula that involves diameter they also told us the height notice that that six inches is the height of the cylinder and this 2.5 is the die so again we've got lateral area and and we are going to find lateral area here because we don't the area the bases doesn't really factor into this notice that the sides of the cylinder are never gonna paint the floor we've got height of six inches in diameter of 2.5 so let's look at our formula okay lateral area is pi times times so we do pi times 2.5 times 6 that gives us 15 pi now would just leave my hand so right there but the book does say that's approximately forty seven point one inches squared so in one full time the stencil roller does cover about forty seven point one inches squared now if you are confused at all about why we used lateral area and not surface area in this problem encourage you to think about why we wouldn't want to include this in the area that the roller paints but also would follow up maybe with question to your classmates or to me so your homework tonight is gonna be number 87 this one should be posted on the Google classroom by now and want you guys to do page 1606 numbers 11 and 15 page 1607 number 18 and you should also check out the 11.3 slides for next time notice again just three problems so shouldn't take you too long and there are some other exercises there if you want extra practice good luck