Volume of a Cone and Pyramid How to Find Formula

its volume equals one-third the area of the base times the height so when you see that capital it means the area of the bottom times the height now you don't want to get confused when you do surface area you're using the slant height you can it's usually used the letter for the slant height that's the angled height or the leaning height here we want the perpendicular height the perpendicular distance from the center of the base to the vertex here that's the overall height okay same thing with the cone we're going from the center of the base okay perpendicular to the the vertex right there okay so let's go ahead and do this example of the pyramid now one thing wanted to show you is that if you put this pyramid okay in box okay like this okay can you visualize this here if you put it in box this pyramid would only take up third of the volume of that box so if the bases were the same the height was the same because of the pyramid is like you know on diagonal it's angled up okay like this you could actually fit two more of these pyramids in there granted you have to cut them up or you know divide them up somehow so that they would you know fit in all those empty spaces there so this is only taking up third that's where you can see the thirds coming from there but let's go ahead and do this so it's volume equals one-third okay area of the base now the base here you can see is square so it's going to be eight times eight so that's 64 times the overall height so we don't want this height here sometimes we'll give you this okay we don't want that one we want 10 and so there you go so it's one-third times 64 times 10 that's going to give your volume and that's going to be in units cubed okay the next one is cone okay so cone it's exact same formula like to kind of group my formulas together like this so there's less to memorize and you might want to do the same thing too so cones and pyramids kind of have the same shape right just different kind of base so you just have to think okay this is circle what's the area of circle pi squared okay so that's the base right there times the height and we want this perpendicular height not the slant height okay the radius is 5 so we have pi times 5 squared so that's pi times 25 times 1 3 times the height which is 10. so if you multiply all that together that's going to give you your volume in units cubed now just quick note sometimes students get little bit you know tripped up on the fractions you can think of any whole number as fraction by putting it over one so you would multiply all the numerators all the denominators and then you would divide so in this case that's 25 times 10 is 250 times pi so that's 250 pi if we want an exact answer if you want to round an answer just use 3.14 for pi and then 3 times 1 times 1 times 1 is 3 and there you go that's an exact answer but again you could make pi 3.14 divide get decimal and round but this is exact