Volume of Pyramids and Cones High School Geometry Lesson

hey everyone in this video we are going to be learning about how to calculate the volume of pyramids and cones first let's start off with some notes so pyramid is 3D shape that has one base and that one base could be variety of different polygons it could be square it could be triangle rectangle and so on whereas cone has one base as well but that base is going to be circle we're only going to work with right circular cones here the volume formula for pyramid is one-third capital times that capital stands for area of the base it is important that that's capital and is the height now since cone is very similar to pyramid it's also the same volume formula however since all the bases of the cones we're going to see are circles instead of using capital we could replace it with pi squared is basically capital area of the base is pi squared whenever the base is circle all right so let's grab our calculators and let's start to look at some sample questions and actually before we do that just one quick note it says here both pyramids and cones have vertical height and slant height if given the slant height use Pythagorean theorem to find the vertical height so we'll look at that in our sample questions all right so let's grab calculators and we are going to need them in order to calculate our volumes so let's start with one here find the volume of the rectangular pyramid so I'm going to start by just writing down what my volume formula is the key is that it says it's rectangle rectangular pyramid that tells me the base is rectangle and to find the volume of rectangle I'm going to do length times width so 8 times 3 in this case the height of the pyramid is 12. so I'm going to put this into my calculator so 1 3 times 8 times 3 times 12. and got that the volume of the pyramid is 96. whenever we find volume our units are cubed so centimeters cubed number two find the volume of the cone in terms of Pi so I'm going to use our shortcut formula for finding volume of cones the one that has replaced capital with pi squared and now can just substitute in our values we can see the radius is 5 and we can see that the height is 9. now since this problem says in terms of Pi what that means is we want pi to be in our answer and that means we do not want it typed into our calculator so I'm going to type in my calculator everything besides the pi okay so the one-third gets typed in the 5 squared then 9 and get 75 but our answer since it's in terms of Pi will be 75 Pi centimeters cubed the reason this is done is so that we don't have to round if you think about Pi it's an irrational number you're going to get decimal every time this gives us an exact volume of the column okay number three hexagonal pyramid has base area of 42 inches squared and height of 6 inches find the volume of the pyramid so our volume formula again 1 3 capital times we don't need to know how to find the area of the hexagon here it just tells us that the area is 42. and the height of the pyramid is six so we just now have to type that into our calculator so 1 3 times 42 times 6 we get 84 and our units are inches cubed number four the square pyramid shown below has an altitude of nine and base with edges of seven inches find the volume of the pyramid here's our formula again area of the base well this is square base know that because it says square pyramid so all the sides are seven so would do seven times seven in order to find the area and the height of the pyramid is nine once we have all of that filled in again we go to our calculator type it in and get the volume of the pyramid is 147 147 inches cubed number five find the volume of right circular cone with diameter of six keep in mind that means our radius is three and height of 14 inches rounded to the nearest hundredth of cubic inch we're going to use that shortcut formula of the one-third pi squared for our cone the radius is three height is 14. we are putting Pi in the calculator because it does not say in terms of Pi anytime you're going to round that's when you are going to put Pi in the calculator so 1 3 times pi times 3 squared times 14 and we're rounding to the nearest hundredth so two places past the decimal that gives me 131.95 cubic inches number six the isosceles triangle shown below has base of 12 inches and height of 5 inches if the triangle is continuously rotated about its altitude what is the volume of the 3D object formed by this rotation in terms of Pi when you're forming solid of Revolution which is what we're doing in this case the way you would draw it out is almost by thinking of reflection if take and reflect it over the altitude in this case it's going to land on if take and reflect it over the altitude it's going to land on and what's going to happen is it's going to basically just map onto itself now we know we need this to be three-dimensional shape so whatever Edge is perpendicular to the axis of rotation that would be AC we're going to curve it and now you can see that this resembles cone so let's use our cone formula 1 3 Pi the radius is 6. notice how the diameter of that cone is 12 so the radius is 6. the height is 5. we're looking for this in terms of Pi so I'm not putting Pi in my calculator I'm putting everything else and get that the volume is 60 pi cubic inches number seven right circular cone has volume of 12.5 Pi meters cubed diameter of 5. keep in mind radius is 2.5 then find the height in meters all right so this one's little bit reverse so what we're doing in this case is we're actually given the volume so I'm going to plug that in for and we're asked to find one of the dimensions so in this case we are going to be solving for what would suggest doing is if that volume is in terms of Pi which it is here get rid of Pi first it's going to cancel out on both sides when you divide by it because you'll get pi over Pi which is 1. so I'm just going to get rid of it right to start and I'm also going to get rid of the fraction the way I'm going to do that is by multiplying both sides by the reciprocal so 3 times 12.5 is 37.5 over on the right the one third and the 3 will go away and then I'll simplify the 2.5 squared now you can see we have much simpler equation to solve divide both sides by 6.25 so 37.5 divided by 6.25 and we get 6 meters as our height of the cone all right last problem here the pyramid shown below has rectangular base with Dimensions 12 by 2.5 if the slant height of the pyramid is 10 what is the volume of the pyramid so mentioned before that if you're given slant height you're going to need to use Pythagorean theorem so here's my right triangle if it helps draw it off to the side the hypotenuse is 10. the bottom leg is 6 because notice it's half of 12 and we're looking for this missing leg here or the height of the pyramid and could use Pythagorean theorem or know my Pythagorean triples and I'm going to get that that value is 8. now that have that can go about my Formula 1 3 capital times area of the base is going to be 2.5 times 12 since it's rectangle length times width the height is now eight and we are going to put that in our calculator so have one third times 2.5 times 12. times eight and we get the volume of this pyramid is 80 inches cubed hopefully this video helps you understand how to calculate the volume of pyramids and cones