Volumes of cones intuition Solid geometry High school geometry Khan Academy

so have two different three-dimensional figures here have pyramid here on the left and have cone here on the right and we know few things about these two figures first of all they have the exact same height so this length right over here is and this length right over here going from the peak to the center of the base here is as well we also know that the area of the bases is the same so for example in this left pyramid the area of the base would be times and let's just assume that it is square so times so the area here is going to be equal to squared and the area of the base so that's area of this base is equal to squared and the area of this base right over here would be equal to area is equal to pi times squared and i'm saying that these two things are the same so we also know that squared is equal to pi squared now my question to you is do these two figures have the same volume or is it different and if they are different which one has larger volume pause this video and try to think about that all right now let's do this together now given that we're talking about two figures that have the same height and at least the area of the base is the same you might be thinking that cavalry's principle might be useful and just reminder of what that is cavallari's principle tells us that if you have two figures and we're thinking in three dimension three-dimensional version of cavalry's principle if you have two figures that have the same height and at any point along that height the cross-sectional area is the same then the figures have the same volume so what we need to do is figure out is it true that at any point in this height do these figures have the same cross-sectional area well to do to think about that let's pick an arbitrary point along this height and just for simplicity let's pick halfway along the height although we could do this analysis at any point along the height so halfway along the height there halfway along the height there so this distance right over here that would be over two this distance right over here would be over two this whole thing is and what we can do is construct what look like similar triangles and we can even prove it to ourselves that these are similar triangles so let me construct them right over here and the reason why we know they're similar is that this line is going to be parallel to this line and that this line is parallel to that line to that radius and how do we know that well we're taking cross sectional areas that are parallel to the base that are parallel to the surface on which it sits in this situation so in either case these these cross sections are going to be parallel so these lines which sit in these cross sections or sit on the base and sit in the cross section have to be parallel as well well because these are parallel lines this angle is congruent to that angle this angle is congruent to this angle because these are transversals across parallel lines and these are just corresponding angles and of course they share this angle in common and here you see very clearly right angle right angle this angle is congruent to that angle and then both triangles share that and so the smaller triangle in either case is similar to the larger triangle and what that helps us realize is that the ratio between corresponding sides is going to be the same so if this side is over 2 and the entire height is so this is half of the entire height that tells us that this side is going to be half of so this right over here is going to be over 2. and this side over here by the same argument is going to be over 2. and so what's the cross-sectional area here well it's going to be over 2 squared so it's going to be over 2 squared which is equal to squared over 4 which is 1 4 of the bases area which is equal to 1 4 of the bases area and what about over here well this cross sectional area is going to be pi times over 2 squared which is the same thing as pi squared over 4 or we could say that is 1 4 pi squared which is the same thing as 1 4 of the area of the base the area of the base is pi squared now we're saying 1 4 pi squared so this is going to be equal to 1 4 the area and we already said that these areas are the same and so we've just seen that the cross sectional area at that point of the height of both of these figures is the same and you could do that one-fourth along the height three-fourths along the height you're going to get the same exact analysis you're going to have two similar triangles and you're going to see that you have the same areas same cross-sectional areas at that point of the height and so therefore we see by cavalieri's principles in principle in three dimensions that these two figures have the same volume and what's interesting about that is it allows us to take the formula which we've proven and gotten the intuition for in other videos for the volume of pyramid we've learned that the volume of pyramid is equal to one-third times base times height and say well this one must have the exact same volume it must also be volumes equal to one-third times the area of the base times the height because in both in both of these cases the area of the base is the same and the height is the same and we know that they have the same volume