Why 1 3 in Volume of Cone

an interesting question is why do we have one-third in front of the volume formula for cone now if you've ever wondered this and you started to search around you most likely would have run into proof which shows you the volume formula through integration and what happens is one can take this comb you can kind of flip it around create an axis so and you can see that same cone it's just flipped and it's touching the Apex is at the origin there and you can integrate because you can just take the actual base and then just shift over so you would have just these slices which you would sum up that's what integration is now what if you're interested in trying to get some intuition of this one-third but you don't really want to integrate so how would you do that so I'm going to remove this for moment so that we don't use this because don't want to use integration to try to convince you that it is indeed one-third now the convincing is going to take an assumption and the assumption is that you are familiar with the volume of pyramid and this just can be any pyramid with particular base now for our purposes we just actually need the base to be regular polygon we don't need it to be anything else now just recall on the volume of pyramid so if you do take any pyramid that you have let me just draw one quickly so here's our pyramid now I've inscribed it in prism can remove that prism and you can then see the pyramid and the formula for this is indeed one-third the base so this is the base area that we have so let's just call this base and multiplied by the actual height so this would have been the height of the pyramid now if you are convinced that the Pyramid has this particular volume then you can indeed intuitively think about the cone as well now if you do need convincing have done very short intuitive video can put up the link to that up above there and for anyone who's really interested in trying to get the general proof which is much longer but think it's worthwhile taking look at that for the volume of pyramid for any base that you would like I'll put link to that as well now once we know this then transferring from here to here is actually not that bad now how can you think about this without any integration well you can think about it in this particular way notice that really the one key item here which changes is that the cone can also be inscribed in an actual cylinder and we know that the cylinder volume would be just the base times the height and The Cone itself is one-third why well because if you take the base which is circle and now you start taking any polygon so notice I'm going to start regular polygons in here so I've kind of Drew this ahead of time you can notice so now this is the top view so the cone base is the circle and then have polygon in here so for instance would have this okay this would have been my first polygon so it is just base with triangle and we know what the volume of that would be and it's inscribed inside now if these are regular polygons so the sides are all equivalent you'll notice that now if extend this to square notice that it covers more of the circle now if extend this to pentagon it's going to be now inscribing even more and now you can imagine this process continuing on could take now four sides five sides six sides and so on but I'm still dealing with polygon now eventually this is going to become very large but now what happens as this becomes larger and larger and larger well what happens is that polygon as so the number of sides approaches Infinity as it goes to really large number it approximates our actual Circle and if we know that that the base can be approximated so the actual base of circle can be approximated by actual polygons if you take an infinite number of size with them and you know what the volume of polygon is it has one-third times the base multiplied by its height well then you're done now you're convinced hope that you can indeed have cone and the volume and that one-third makes sense because it's just an extension okay of the actual pyramid where the base okay starts to grow in the number of sides that it has now I've actually encoded this for you and here it is I'm using decimals to do this but wrote the code in on the side and you can't see all the code in here but what I've done is I've actually inscribed circle and then now what I'm going to do is I'm going to hit play on this and you're going to see because don't want to be drawing all of these well you can hopefully be convinced that as the polygon increases in the number of sides so notice here it's actually only at 22 23 24 and so on so it's very little but if you make this larger and larger and larger so more sides you can now clearly see even with this illustration that the polygon approximates circle now this is from the interior and you can just keep going now on the left hand side of also shown you the actual areas so notice you have an area of your circle which is pi squared now assume the is equal to one does it matter what it would be but one is convenient but it again it is is irrelevant and notice that just below it there is the area of the polygon okay as it's increasing now notice what it's tending to and at the bottom which is the area of the circle minus the area of the polygon notice that the difference between them just gets smaller and smaller and smaller why because the polygons okay as tends to Infinity basically create circle for us and so from that we can actually conclude that indeed intuitively this one-third for the cone completely makes sense it's just an extension of pyramids that's all that it is now to be completely honest with you this still uses some part of calculus because it uses limits as tends towards Infinity is basically limit and if you really want it to show this you can't just show interior that it encapsulates and basically approximates circle you also would have to do the same thing from the exterior so meaning as you can see here you can now create bigger triangle and that triangle encapsulates the circle inside then it goes into well square that square encapsulates the circle but look what happens then it gets into pentagon and notice that our polygon just gets smaller and smaller and it gets closer and closer to the circle and in the illustration that did in the simulation didn't do both of them but did the one on the interior where it approximates it from the inside and it also will approximate it from the outside which is going to engulf and basically have circle now this is particular theorem actually used from limits but you don't need to know it hopefully this convinces you that indeed the area that we have or the base itself can be approximated by polygons and we know what the volume of pyramid is with base of polygon and there you have it that's why we have one third of the actual cylinder existing in our formula intuitively without any integration but hopefully convincing visual representation for you thanks for watching okay give it thumbs up if you liked it we'll see you in future videos bye welcome to 1 million journey.com cheers