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CCSS.Math: ,

let's find the volume of a few more solid figures and then Maggie if we have time we might be able to do some surface area problems so let me draw a cylinder over here so that is the top of my cylinder and then this is the height of my cylinder this is the bottom right over here if this was transparent maybe you could see the backside of the cylinder so you can imagine this is kind of looks like a soda can and let's say that the height of my cylinder H is equal to 8 I'll get some units 8 centimeters that is my height and then let's say that the radius of one of these of the top of my cylinder of my soda can let's say that this radius over here is equal to 4 centimeters so what is the volume here what is the volume going to be and the idea here is really the exact same thing that we saw in some of the previous problems if you can find the surface area of one side and then figure out kind of how deep it goes you'll be able to figure out the volume so we're going to do here is figure out the surface area of the top of this cylinder or the top of this soda can and then we're going to multiply it by its height and that will give us a volume this will tell us essentially how many square centimeters fit in this top and then if we know if we multiply that how many by how many centimeters we go down then that will give us the number of cubic centimeters in this cylinder or soda can so how do we figure out this area up here well the area of the top this is just finding the area of a circle you could imagine drawing it like this if we were to just look it straight if we were just look at it straight on that's a circle with a radius of 4 centimeters the area of a circle with radius 4 centimeters area is equal to PI R squared so it's going to be pi times the radius squared times 4 centimeters squared which is equal to 4 squared is 16 times pi and our units now we're going to be centimeters squared centimeters squared or another way to think of these are square centimeters so that's the area the volume is going to be this area times the height so the volume is going to be equal to sixteen pi centimeters squared centimeters squared times the height times eight centimeters times eight centimeters eight centimeters and so when you do multiplication you can the associative property you can kind of rearrange these things and the commutative property we doesn't matter how what order you do if it's all multiplication so this is the same thing as 16 times eight let's see 8 times 8 is 64 16 times 8 is twice that so it's going to be 128 pi and you have centimeters squared times the centimeter so that gives us centimeters centimeters cubed or 128 PI cubic centimeters remember pi is just a number we write it as pi because it's kind of a crazy irrational number that never that if you were to write it you couldn't even you know you could never completely write pi 3.14159 keeps going on never a repeat so we just leave it as pi but if you wanted to figure it out you can get a calculator and this would be 3.14 roughly times 128 so it would be like you know close to close to 400 cubic centimeters now how would we find the surface area how would we find the surface area of this figure over here well part of the surface area the to surface is the top at the bottom so that is the that would be part of the surface area and then the bottom over here would also be part of the surface area so if we're trying to find the surface area let's do surface let's find the surface area of our cylinder it's definitely going to have both of these areas here so it's going to have the 16 pi centimeter squared twice this is 16 pi this is 16 PI square centimeters so it's going to have 2 times 16 PI centimeters squared I'll keep the unit's still so that covers the top and the bottom of our soda can and now we have to figure out the surface area of this thing that goes around and the way I imagine it is imagine if you were trying to wrap this thing with wrapping paper so let me just draw let me just draw a little dotted line here so imagine if you were to cut it just like that cut the side of the soda can and if you were to kind of unwind if you were to unwind this this thing that goes around it what would you have well you would have something you would end up with a sheet of paper where this length right over here this length right over here is the same thing as this length over here and then it would be completely unwound and then these two ends let me do this in magenta these two ends these two ends use to touch each other and I'm gonna do it in a color that I haven't used yet I'll do it in pink these two ends use to touch each other when it was all rolled together and they used to touch each other right over there so the length of this side and that side is going to be the same thing as the height as the height of my cylinder so this is going to be 8 centimeters and then this over here is also going to be 8 centimeters and so the question we need to ask ourselves is is what is going to be what is going to be this this dimension right over here how far and remember that dimension is essentially how far did we go around around the cylinder well if you think about it that's going to be the exact same thing as the circumference of either the top or the bottom of the cylinder so what is the circumference the circumference of this circle right over here which is the same thing as the circumference of that circle over there it is 2 times the radius times pi or 2 pi times the radius 2 pi times 4 centimeters which is equal to 8 pi 8 pi centimeters so this distance right over here is a circumference of either the top or the bottom of the cylinder it's going to be 8 pi centimeters so if you want to find the surface area of just the wrapping just the part that goes around the cylinder not the top or the bottom it's going to when you unwind it it's going to look like this rectangle and so it's area the area of just that part is going to be equal to 8 centimeters times 8 pi centimeters so let me do it this way it's going to be 8 centimeters times 8 pi centimeters 8 pi centimeters and that's equal to 64 pi 8 times 8 is 64 you have your pi centimeters centimeters so one you want the surface area of the whole thing you have the top you have the bottom we already threw those there and then you want to find the area of the thing around we just figure that out so it's going to be plus 64 PI centimeters squared and now we just have to calculate it so this gives us 2 times 16 pi is going to be equal to 32 that is 32 PI centimeters squared plus 64 pi 64 pi let me scroll over to the right a little bit plus 64 PI centimeters squared and then 32 plus 64 is 96 by 96 PI centimeters squared so it's equal to 96 PI square centimeters which is going to be a little bit over 300 square centimeters and notice when we did surface area we got our answer in terms of square centimeters that makes sense because surface area it's a two dimensional measurement we think about how many square centimeters can we fit on the surface of the cylinder when we did the volume we got cubed or we get centimeters cubed or cubic centimeters and that's because we're trying to calculate how many how many one by one centimeter cubes one by one by one centimeter cubes can we fit inside of this structure and so that's why it's cubic centimeters anyway hopefully that clarifies things up a little bit