Cavalieri's Principle and composite solids

welcome back to boom your math today we're going to look at the volume of prisms or we're looking at the volume prisms and cylinders but specifically at cavaliers principle and how to find the volume of composite figures okay so let's start with a quick review find the area of each figure round to the nearest tenth you need to be know this stuff for um maybe a test or a quiz coming up okay an equilateral triangle has an edge of 20. so we're going to try to find the area okay so we've got an equilateral triangle with 20 on each side how do you find the area of that well it's one half base times height correct okay so we got the base that's 20 but what's our height hmm let's draw it in okay on an equilateral triangle what happens to this base it cuts it in half so we end up with 10 over here and 10 over here well once you get that what kind of triangle is this left side that's a 30 60 90 triangle so if that's a 30 60 90 triangle this angle was 60 but i cut it in half so that's my 30. that means this side that's 10 is my short leg this 20 is my hypotenuse so i need to know the long leg short leg to long leg what do we multiply by we multiply by root 3 and there's our height so we're going to have one half of 20 times oops 10 root 3. now i started writing 10 root 10 but it's 10 root 3 so half of 20 is 10 and 10 times 10 is 100 root 3 and that would be the area and that's in centimeters squared right because it's area okay so that's how you do that one do be familiar with that one find uh the area of a regular hexagon well we haven't done hexagons i don't think in this this lesson so don't worry about the hexagon a circle with a radius of 6.8 okay what's the area of a circle pi r squared i hope you're thinking okay so if the radius is given it's very easy which we just take pi times 6.8 quantity squared so let's grab our calculators and this would be one you'd use the calculator on 6.8 squared would give us 46.24 so 46.24 pi and that would be inches squared so that one's pretty easy too and then a circle with a diameter of seven now remember we're dealing with circles again and the formula is pi r squared so if our diameter is 14 what's our radius our radius is seven so pi times seven squared or 49 pi feet squared so there's a quick little review and again i don't think we're going to do hexagons this summer we just don't have enough time okay so let's talk about cavaliers principle okay the volume of three dimensional figures is the number of non-overlapping unit cubes of a given size that will fit exactly on the interior so we're talking about the inside space so you can see this pyramid our prism and then you can see this prism well they're the exact same thing only one is slanted we also have another word for that oblique is another word and cavaliers principle says that if these two two figures have the same height so top to bottom top to bottom same height and they have the same cross section so their bases are the same size then they have the same volume so really a slanted one you've treat exactly like one that's straight up and down you just have to be careful to find this height okay now let's put this to practice here oh so down here you've got the formulas the volume of the prism with a base of b big base area b and a height of h is volume equals b big b times h and then the length width and height right our base area times our height and then when it's a cube side squared times height but they're just going to be oblique sometimes so that's where just cavaliers principle is just saying the oblique ones are calculated the same way so here's an example of one with cds and it's really a great example or coins are another good way if you stack up like you know you have two stacks of cds let's say you have 15 and this one at 15 and this one and this one slants does the volume change no it's the same volume it's just slanted so there's kind of proof of how to do that one and remember the volume is base area which in this case cds would be pi r squared times the height so pi r squared times a height but notice on the oblique one the height is that altitude from the top corner down so okay let's put it into practice here so we're going to find the volume of this oblique cylinder so we need to know pi r squared h right pi r squared h is our formula so our radius what's our radius here it's nine so nine squared and what's our height well that's given to us that was really easy 14. sometimes that won't be given you might be given this and you might have to work a pythagorean theorem to figure out your height so always keep pythagorean theorem in your back pocket okay so let's go back and calculate well we know 9 squared is 81 and then times 14 i don't know that off the top of my head so it's 11 34. so that's 1134 pi and this is a volume so it's inches cubed three-dimensional so there you go there's our first one with an oblique okay so find we're going to do one without a picture find the volume of a cylinder that has a base area of 121 pi centimeters squared and a height equal to twice the radius find the volume in terms of pi so if we know the base area is 121 pi that was given to us what would be the radius well base area of a circle is pi r squared and that's equal to 121 pi so let's solve for r well let's divide both sides by pi right that would just cancel pi off of each side since they have the same thing so r squared is 121. so what's our radius our radius is square root of 121 or 11. okay now it says that the height is equal to twice the radius so what's twice the radius well the radius is 11 so the height must be 22 2 times 11 or 22. so now we already have the base area so that's this is going to be easy now base area times height base area is 121 pi times a height of 22 so let's grab our calculators 121 times 22 and that's 2 6 2 2 62 pi don't forget your pi leave your in terms of pi and then this would be in centimeters and it's volume so it is cubed okay so we that was a working backwards one which we're gonna have to do a few of those to kind of work backwards towards okay this sheet just has some answers on it but let's keep going okay um let's look at this oblique and let's talk about what this is called well the bottom of it is what shape it's a rectangle so this is a rectangular prism but is oblique so it's an oblique rectangular prism okay so we need to find the base area don't we so the base area is just going to be length times width which is 4 times 3 and then the height the height is not the slanted part but the straight up and down part which is 1.5 so let's see um the volume would be 4 times 3 times 1.5 okay so the volume is 18 and this would be feet cubed okay now what happens if we double all the dimensions so what are the dimensions if they're multiplied by two well this side is no longer four it's eight this slot side is no longer three it is six and this height is no longer one and a half but it's three okay so we've taken each side and multiplied it by two haven't we okay what do you think it's gonna do to our volume is it gonna double it is it gonna triple it what's it gonna do well let's just crank it out for a second plug into your formula a length times a width eight times six times your height which is three and that's 48 times three let's see 48 times three should know that one 144 wow it's a lot bigger isn't it it's not twice as big it's not four times as big how many times bigger is it well 144 divided by 18. it's eight times larger so it's eight times larger why would it be eight times larger and not doubled well we did a length and a width and a height all by two so what's two times two times two ah it's eight so it increased each dimension three times so it's three times that dimension cubed and that's gonna come into play in just a moment okay so let's keep going um let's look at composite figures okay composite figures the volume of a rectangular prism oh it says find the volume of the composite figure round to the nearest tenth so what kind of figures do we have well at first you might say ah that's a pyramid and a prism on the bottom but be careful the top is also a prism because it's got parallel bases but the bases just happen to be triangles so what you're going to do is you're going to find the area of the rectangular prism on the bottom and we know that one's easy it's length times width times height so let's do that length times width times height so that's going to be 8 times 4 times 5 and that's going to be 40 times 4 which is 160 okay so 160 cubic centimeters now let's find the volume of the regular triangular prism this triangle right here okay do you see how they're parallel to each other and that's how you can tell okay so we've got to find the area of an equilateral triangle don't we oh we did that in the warm-up so let's draw our altitude down we know that cuts that into four and four and remember the rule this is a short leg this is the long leg so that is for square root three so let's find the the area is one half the base well we're going to use the whole base of the triangle one half of eight times the height which is four root three so half of eight is four and four times four is 16 root three so that is now my base area so base area times the height well now the height is this part right because here's my parallel side so times five and what's 16 times five i believe that's 80 isn't it okay yeah 80. so that's going to be 80 roots 3 cubic centimeters okay now here are my two volumes i'm putting them together so i'm going to add them now can i add these together not in the form they're in so there's your answer in exact form now if you have a decimal form you can go ahead and type it out as 160 plus 80 and put in the square root 3 and it should give you the decimal answer so the decimal answer is 298.6 cubic centimeters but if you want the exact because we'd have to round this off so that one's not exact here's the exact answer so if you're asked for exact answers leave them as radicals and if you're asking ask for estimated you write it as a decimal okay let's look at a couple more of these composite figures okay this next composite figure is a rectangular prism topped off with a cylinder but what do you notice about the cylinder the cylinder is only half of a cylinder so let's look at this rectangular prism and then we're going to figure out that cylinder part so the rectangular prism is going to be length times width times height or 9 times 11 times 6. so 99 times six is five hundred and what something almost six hundred uh four six short of four hundred right five ninety four cubic millimeters okay so that part's either the rectangular prism now let's look at the cylinder okay cylinders volume is pi r squared times height but what are we going to do with that cylinder since we only have half of it we're going to divide it by two okay so let's go back and figure out our radius and our height okay what's the radius of this cylinder hmm it's half of this side isn't it okay so this is nine and this is nine so it's going to be four and a half so it's going to be 4.5 squared and now the height is this height which is 11. then we're going to take all that divided by two so let's crank this out let's see 4.5 squared that's going to give me 20.25 times 11. that's going to give me 200 and 222.75 pi divided by two so let's just divide that coefficient by two and that's a hundred and eleven point three seven five and i still have my pi in there okay so that is the volume of that cylinder part now we've got to take these two and do what add them together so i've got 594 plus 111.375 pi now we usually round to tenths so i'm going to round that three to what a four so 594 plus 111.4 pi and because that has a pi i can't add them together and i just stick my units with it which is millimeters cubed and there we go okay now let's look at composite figures where we're got a hole in the middle okay so we've got a cylinder and in the middle of it stuck cut out of the center of it is this prism so instead of adding the two quantities together like we did up here for the composite figures we're going to do what we're going to subtract because we'll take the volume of the cylinder or cutting out the volume of that rectangular or i'm sorry that's actually a square a square prism so let's first find the volume of the cylinder volume of the cylinder is pi r squared base area times height so our radius is easy it's given us 3 so that's going to be 9 pi times 5 or 45 pi okay now i need to find the volume of the square prism i'm not given a side of the square i'm given this little measurement and that surely is not the side so let's look at this from looking straight down i'm changing this to a bird's eye view so this is now three okay which means this is three okay what do we know about a square these are right angles and when we draw this in it's going to become a 45 45 90 right triangle so that means this side is the hypotenuse of the 45 45 90. so it's three root remember the rule three root two you multiply the leg by root two so that's now the side so let's see the volume of a square prism is going to be base area which is side squared times the height and then my side is 3 root 2 squared times the height well the height is still 5 and so 3 squared is 9 and square root of 2 squared is just two so that's going to give me 18 times five what's 18 times five gives me 90. so that's going to be 90. so the volume of that square prism is 90. now what do i do with these two since one's cut out instead of adding this time i am going to subtract them so it's going to be 45 pi minus 90. now you may think oh 45 is smaller than 90 how could you subtract this but watch 45 times 3.14 it is not less than 90. it's 141.3 minus 90. 51.3 if you were asked for the decimal if you're not asked for if you're asked for exact you stop right here and just throw your units in there as centimeters cubed if you're asked for an estimate or decimal you would round it off to this point so there you go there's one of those where you're subtracting composites and here's a quick little i printed this out for myself to make sure i made i did it correctly and then the last one i want to do with you is this composite yeah this is the last one right here is the composite of a cylinder inside a cylinder so it's like a washer it's like a stack of washers or stack of cds same concept you're going to do the volume of the large one and subtract the volume of the small one but this time they're both circles so it's going to make the ansi answer easier okay so we got a circle minus a circle so think about your shapes what are you subtracting so pi r squared is my formula times the height minus pi r squared times height so the formulas are the same the only difference is going to be what the heights are the same but the radiuses are different so what's the radius of the larger one well that's 18 so it's going to be 18 squared times 12 minus pi and now my radius of the small one is 9 so 9 squared times twelve and let's crank this out okay eighteen squared times twelve that's going to give me three thousand eight hundred and eighty eight pi minus okay 9 squared is 81 and 81 times 12 is 972 pi so now i'm going to subtract these can i subtract these and the answer i hope you're saying is yes you can because they both have a pie with them so three hundred and eight or three thousand eight hundred and eighty eight minus 972 is 2916 pi don't forget your pi inches cubed and that's how you find the volume of composite figures when one is inside the other so when they're stacked on top of each other all you add when one's inside the other taking out its space you subtract so hope this video was helpful