Math 7 Geometry II Unit Lesson 3 Surface Area of Rectangular Prisms

hey guys welcome back we are gonna talk about something called surface area today and we are gonna find surface area of all different three-dimensional shapes in this unit but today what we're gonna do is we're gonna focus on rectangular prisms so want you to think of rectangular prism like box okay so don't know about you but I've been getting lots of Amazon packages in the mail during this quarantine and so if you think about those boxes whoever is creating these boxes needs to figure out how much cardboard is needed to make them and so what they're really calculating is the surface area so want you think of surface area as they're covering something okay so when you wrap present you're covering whatever it is that you've purchased and so you're trying to figure out how much wrapping paper do need to cover that that surface area so anything where you're covering three-dimensional object is considered surface area and remember area is two-dimensional shape how much space is inside is inside that two-dimensional shape where surface area is how much area is needed to cover three-dimensional shape okay so that's the main difference between area and surface area two-dimensional versus three-dimensional so like said we're gonna look at rectangular prisms today so what is rectangular prism said it was box but really what it means is that all of the faces face meaning side so all of the sides are rectangles okay another type of rectangular prism is cube and this would be something where all of the faces or all of the sides are squares okay so you guys all know what cubes are think about die you know six-sided die that's cube all the sides all the faces are squares okay so in order to find the surface area of rectangular prisms what we're gonna do is we're going to find the area of each of the faces or each of the sides and when we're done doing that we'll add all the areas together and that will give us our surface area okay so if you look at example one you can see that we have this rectangular prism where the length is 8 the width is 3 and the height is 6 so our job in order to find the surface area is to find the area of each of the faces each of the sides so any rectangular prism is going to have six sides there's top bottom front back and left and right the good news is any sides that are opposite meaning the top and bottom the left and the right the front in the back they will always have the same area so really what you're doing is you're finding the area of three sides and then you're just doubling it okay which is kind of nice so if you look at this again in example one the top I'm gonna shade in the top because think that'll help you visualize the side that I'm talking about okay so here's the top of that rectangular prism and so because it's rectangular prism we know all the sides are rectangles and to find the area of rectangle we're gonna do area equals length times width okay so for each face we're doing length times width so to find the area of the top need to figure out what is the length and what is the width well because know that this is eight that means that this side is also eight and because know this is three know that this side is also three so the area of the top would just be eight times three or 24 okay so then the area of the bottom is the same okay so the top and the bottom always have the same area we can see that because here's the bottom okay so that we can see as 8x3 also which is 24 okay so now let's do the front in the back me erase all this you don't have any there okay so the front of the box so this is considered to be the front of our box and again it's rectangle so I'm going to do the length times the West so can see the length of the front is 8 what don't have is the width but do know that this side is 6 which means so is this side and so there's our length and width so I'm going to do 8 times 6 which is 48 ok now the back has that same area let me show you the back just so that we're clear so this is the back side here little trick on your eyes so the back has the same length as the front so this is 8 we can see the width there is 6 which is what makes it the same 8 by 6 is 48 all right so the last two sides we have are the left and right I'm actually gonna do the right side first because here's the right side know the length and the width 6 and 3 know that's the same side which is over here both of those are 6 times 3 which is 18 okay so our last step is to add all of those areas together ok so the surface area is the sum of all the areas of the faces okay you don't have to write that out every time I'm just showing you so that you know what I'm adding when you add all 6 sides together you should get 180 now for label it's still area so we're going to use meter squared okay so surface area is still area so still in square all right awesome now you guys are gonna love cubes because what do we know about cubes we know all the faces are the same all the faces are squares okay so in this example here we can see that every side has length of four okay so if start with the top which is right here it's square know it doesn't look like it but it is because we know it's cube so the area of square is side squared so the area of that top would be four times four is 16 which is the same as the area of the bottom also 16 okay so what's really cool about cubes is that you really just need to find the area of one of the sides one of the faces you just add that together six times because all six faces have same area all right so let's go ahead and add that 16 plus 16 plus 16 six times or you could just do 16 times six and you're gonna get that the surface area is 96 squared all right let's go back and do another rectangular prism so example three so we're gonna start with the top number we're using like times with just rectangular prism all of the faces are rectangles so here's my top the length and the width four times to the bottom is the same - okay in the back right okay so can see the length of the front seven but the whip have to use is for those two sides are the same another 28 all right left and right she used to do the right numbers here so the right side has length of 7 and up 2 which is 14 the right side is also 14 okay so surface area we're gonna add all of those sides together should get hundred awesome let's do one more cute here so all of the sides have to point one as their length the cube we're doing side squared for each face so if look at the top which is the same at the bottom so I'm just gonna do this can see that it's two point one times two point one which is four point and like we saw in example two in cube all of the faces have the same area so once you find one don't hesitate to just show me all the rest are the same how you got that four point four one the beginning okay so have to do four point four one times six or add it together six times and should be getting twenty six point four six centimeters squared all right hopefully you guys are following along okay with that so we're gonna do two more and what recommend is stopping the video here seeing if you can get number five and six correct I'm home so stop it try it and then replay it and you can watch my work and see if you get it correct okay all right so in these ones didn't list out the six eyes for you I'm just gonna okay so the top and the bottom can see right here's the bottom so that's three by four okay so if you did this one correctly you added up all the sides you should have gotten 164 okay alright let's see how you did with the cube so remember same thing for the cube but what's kind of nice that takes longest all of these sides out you guys can absolutely abbreviate to CB RF m4 step ok so didn't write this over here this one for this one we're doing sides squared so each side has length of 9 so the area of every face is 9 times 9 which is 81 so 81 times 6 is going to give us 486 squared all right so hopefully you guys got those two the last thing we're gonna do is couple word problems okay drawing here smart class so number seven says dog food box has length of six inches width of eight inches and height of 15 inches find the amount of packaging material that will be needed to completely cover the box okay so you know how to draw three-dimensional boxes here's how do it so start with kind of like the front face okay and I'm drawing it like this because it tells me that the height is 15 so know it's gonna be taller than it is wide and then what do is draw another one side like that and then just connect my corners that's my easy way of making three-dimensional box okay so after you have your box drawn would definitely label okay that's hope I'm not drawing it so it tells me the height is 15 inches the length is 6 and the width is 8 now it doesn't matter what call the length and what call the Whitley not so you want to say that this is 6 and this is 8 or reverse that one's 8 that one's 6 it really doesn't matter okay it all will work out trust me all right so let's go ahead and read okay so top and bottom can see those faces are six by eight forty-eight for those two areas the left and the right can see are 15 by 8 and then the front in the back will kind of highlight this one for you so the front is right here if know this is 15 so is this okay so 6 by 15 is my back which is okay so let's add all those up and 516 so find the amount of packaging material that will be needed to completely cover the box 516 square inches of packaging material will be needed you guys will notice if you have roll of wrapping paper you know that you've not opened yet you'll see right on the outside of that wrapping paper it tells you how many square inches or how many square feet that roll will cover so next time you go to buy wrapping paper you're in the store and you see some check it out some rolls cover much more than others it's interesting to see this is definitely where it comes into play all right number 8 Ryan wants to wrap gift that is box measuring 11 by 11 by 11 centimeters how much wrapping paper will he need okay well since we know the box has all three of the same dimensions we know this must be cute okay so try my best to draw cube here eleven so it means the length the width and the height are all 11 centimeters okay so all right so what do we know about pubes we know all of the faces have the same area so once find the area of the top all six faces are going to be that same number all right so I'm actually gonna look at the bottom just because of my picture it's easier to see that that square area every side all right so how much wrapping paper will II need let's see we need to add up 121 six times four 121 times 6 that need 726 square centimeters all right last one leacy uses two hundred and fifty three point five square inches of paper to cover box in the shape of cube what are the lengths of the sides of the cube okay so this is little bit tricky know the surface area they don't come out and say this is the surface area they say this is how many square inches of paper we use to cover the box so we have to know that that number that they gave us is representing the surface area and what they're asking us for is what the side lengths are if this is cube this box is okay so we kinda have to think backwards little bit here if know that all six sides of my cube all the areas of all six sides were added together to get to 53.5 could figure out what the area of one side of that cube would be if know all six sides together is to 53.5 how do find the area of one side you would divide by six right so if take that surface area and divide it by six that's going to tell me with the area of each side of that cuba's so if divide get 42 0.25 squared so again that's the area of each side of my cube okay now I'm not done because it doesn't want that it wants to know what are the lengths of the sides of the cube so let's think about if each side has an area of 40 2.25 how can figure out what the lengths of the sides were well to figure out the area of square which is what the sides of the cubes cube is you would have to say what number times itself gives me 40 2.25 well that sounds like horrible thing to have to guess and check to figure out and so one way that we can do this and we did this with circle is we can take the square root the square root of number tells me what times itself gave me that number so if take the square root of 42.5 I'm going to get 6.5 and so what that means is that each side of that cube must have been 6.5 inches six point five times six point five gives me 42 point two five and then multiply that number by six and got my surface area so that's how you can kind of work backwards to check