Volume of a cone

let's think a little bit about the volume of a cone so a cone would have a circular base where I guess depends on how you want to draw it you think of like a conical hat of some kind it would have a circle as a base and that would come to some point so it looks something like that this you could consider this to be a cone just like that or you can make it upside down if you're thinking of an ice-cream cone so it might look something like that that's the top of it and then it comes down like this this also is kind of those disposable cups of water you might see it's some water coolers and the important things that we need to think about when we want to know about the volume of a cone is we will definitely want to know the radius of the base we definitely want to know the radius of the base so that's the radius of the base or here it's the radius of the top part you definitely want to know that radius and you want to know the height of the cone you want to know the height of the cone so let's call that H right over here you could call this distance right over here H and the formula for the volume of a cone and it's interesting because it's close to the formula for the volume of a cylinder in a very clean way which is somewhat surprising and that's what's neat about a lot of this three-dimensional geometry is that it's not as messy as you would think it would be it is the area of the base well what's the area of the base well that's going to be the area of the base is going to be PI R squared it's going to be PI R squared times the height times the height and if you just multiply the height times PI R squared that would give you the volume of the entire of an entire cylinder that looks something like that so this would give you this entire volume that looks that looks of the figure that looks like this where it's center of the top is the tip right over here so if I just left it as PI R squared H or H times PI R squared so the volume of this entire can this entire cylinder but if you just want the cone it's one-third of that it is one-third of that and that's what I mean when I say it's surprisingly clean that this cone right over here is one-third the volume of this cylinder that is essentially you can kind of think of the cylinder as bounding around it or if you wanted to rewrite this you could write this as 1/3 times pi or PI over 3 times H R squared however you want to view it the easy way I remember it for me the volume of a cylinder is very intuitive you take the area of the base and then you multiply that times the height and so a volume of a cone is just 1/3 of that it's just 1/3 the volume of the bounding cylinder is one way to think about it let's just apply these numbers just to make sure that that it makes sense to us so let's say that this is some type of a conical glass the types that you might see the watercooler and let's say that we're told that it holds so this thing holds 131 cubic centimeters of water and let's say that we're also told that it's height we're told that it's height right over here you only do that in a different color we're told that the height of this cone is 5 centimeters and so given that if what is roughly the radius of the top of this glass let's just say to the nearest tenth of a centimeter well we just once again have to apply the formula the volume which is 131 cubic centimeters is going to be equal to 1/3 times pi times the height which is five centimeters five centimeters times the radius squared times the radius squared or if we wanted to solve for the radius squared we could just divide both sides by all of this business and we would get radius squared radius squared is equal to 131 centimeters school to the third all right 131 cubic centimeters I should say you divide by 1/3 that's the same thing as multiplying by 3 and then of course you're going to divide by PI and you're going to divide by 5 centimeters 5 centimetres now let's see if we can clean this up centimeters will cancel out with one of these centimeters so you left just be left with square centimeters only in the numerator and so this is going to be an N to solve for R we could take the square root of both sides so we could say that our R is going to be equal to the square root of three times one thirty one is 393 over five PI over five PI so that's this part right over here and the square root once again remember we can treat units just like algebraic quantities the square root of centimeter squared well that's just going to be centimeters which is nice because we want our units in centimeters so let's get our calculator out to actually calculate this kind of messy expression turn it on C square root of 393 divided by five times pi five times pi is equal to five I was pretty close so to the nearest it's it's pretty much five centimeters so this is so our radius is approximately equal to five centimeters at least in this example