Cylinder Volume and Surface Area Finding the volume and surface area of a cylinder
Cylinder Volume and Surface Area
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- Let's find the volume of a few more solid figures, and if we have time we may be able to do more 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 would be able to see the backside of my cylinder.
- so you can imagine this kind of looks like a soda can.
- And let's say that the height of my cylinder "h"
- is equal to 8. I'll give some units- 8 cm.
- That is my height.
- And then let's say that the radius of one of these, of the top of my cylinder, or my soda can
- let's say that this radius over here is equal to 4 cm.
- 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 the figure out kind of how deep it goes, you'll be able to figure out the volume.
- So what we are going to do here is figure out the surface area of the top of this cylinder.
- The top of this soda can
- and then we are going to multiply it by it's height and that'll give us a volume.
- this will tell us essentially how many square cm fit into this top
- and then if we know; if we multiply that by how many cm 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 on top
- this is just finding the area of a circle
- you could imagine drawing it like this
- if we were just to look at it straight on
- that's a circle with a radius of 4 cm.
- the area of a circle with an area of 4 cm
- area is equal to pie r squared.
- so it's going to be
- pie times the radius squared
- times four cm squared
- which is equal to
- 4 squared is 16
- times pie
- and our units now
- are going to be cm squared
- or another way to think of these is square cm
- 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
- 16 pie cm squared times the height
- times 8 cm
- times 8 cm and so
- when you do multiplication
- you could use the associative property
- you could kind of rearrange these things
- it doesn't matter what order you do it.
- it is all multiplication
- so this is the same thing as 16 times 8
- let's see 8 times 8 is 64
- 16 times 8 is twice that so it's going to be
- 128 pi, then you have cm squared times cm
- so that gives us cm cubed
- or 128 pi cubic cm.
- remember, pi is just a number
- we write it as pi because it is kind of a crazy irrational number
- that if you were to write it, you could never completely write pi
- 3.14159 keeps going on
- never repeats
- 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 roughtly
- times 128
- so it would be like you know close to 400
- cubic cm
- now, how would we find the surface area of this figure over here
- well, part of the surface area, are the two surfaces, the top and the bottom.
- so 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 are trying to find the surface area
- let's do, surface
- let's find the surface area of our cylinder
- it's deffinitely going to have both of these areas here.
- so it's going to have the 16 pi cm squared twice
- this is 16 pi, this is 16 pi square cm.
- so it's going to have two times 16 pi.
- cm squared.
- i'll keep the units 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 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 used to touch each other
- And let me do a color that I haven't used yet
- I'll do it in pink
- These two ends used 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 of my cylinders
- This is going to be eight centimeters
- And then this over here is also going to be eight centimeters
- And so the question we'd ask ourselves is
- is what is going to be this dimension right over here
- And remember that dimension is essentially how far
- did we go 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 2pi times the radius. 2pi times 4 cm which is equal to 8 pi cm. So this distance right over here
- is the circumference of either the top or the bottom of the cylinder. It's going to be 8 pi cm.
- 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 be unwinded, 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 cm by 8 pi cm.
- So let me does this, it's going to be 8 cm times 8 pi cm. And thats equal to 64 pi. 8 times 8 is 64.
- You have your pi centimeters squared. So when 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 figured that out so it's going to be plus 64 pi cm 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
- cm squared. Plus 64 pi, let me scroll over to the right a little bit. Plus 64 pi cm squared.
- And then 32 plus 64 is 96 pi cm squared. So it's equal to 96 pi square cm is going to be a little over 300 square cm.
- And notice when we did surface area. We got our answer in terms of sq cm. That makes sense because
- surface area is a 2 dimensional measurement. We think about how many sq cm can we fit on the surface
- of the cylinder. We did the volume we got cubed centimeters. And thats because we're trying to figure
- out how many 1x1 cm cubes. 1x1x1 cm cubes can we fit inside of this structure so that's why
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