Volume and Surface Area of Composite Shapes

all right these are the notes that go along with volume and surface area of composite figures example one how many cubic feet of popcorn does this canister hold the first thing that want you to see is cubic feet that should be your hint that you're looking for volume okay so we're going to try and find the volume all right have two shapes have rectangular prism and also have cylinder okay so let's talk about the formula for the volume of rectangular prism the volume of rectangular prism is length time width time height you have that in your notes and to that we need to add the volume of cylinder now remember volume is the area of the base the base of cylinder is circle CLE so need to use pi 2 pi 2 times the height of my cylinder which is okay so let's just plug in some numbers and see what we get for our volume all right so the volume equals 14 * time plus the volume of the cylinder which is 3.14 * 4^ 2quared * 8 the height of my cylinder all right I'll give you second to do some calculations here and hopefully you come up with the volume is 2 ft squared okay so we could actually type out this container holds 1,745 squared all righty perfect that looks much better all right example number two John wants to paint that should be John wants to paint his barn one gallon of paint covers 100 square ft how many gallons of paint will John need all right so when we talk about John painting and we see the words square feet that should tell us that John is going what we're trying to find here is actually going to be surface areas because square feet implies surface area okay so we have square feet and that's talking about surface area so let's see what kind of shapes we have here now remember surface area is the covering but we need to talk about this little bit okay so first of all we can talk about the two shapes that we have for the surface area we have rectangular prism and we have triangular prism okay now the thing is Jon is painting so want you to think about what you actually paint so if this is his barn do you think John's going to paint the bottom of the barn do you think he's actually going to paint the bottom layer this down here do you think that that bottom layer is going to get painted would say probably not just the same as he's not going to going to paint the ceiling because we don't see that we have triangular prism on to top of that so actually the things that John is really going to paint the places that John is going to apply paint would be the ones that highlight right here he'll paint the front of the barn and the back of the barn so we can calculate the area of that and since those two shapes are congruent we can calculate the area of one and add them together he is also going to paint the sides of the barn the right side as well as the left side okay don't think John is going to paint his roof he probably has nice roof and doesn't want to paint that but he probably will paint these front triangular pieces and those are also congruent to one another okay so let's see the surface area is really not an exact formula so we're going to calculate the surface area of the front and the back so have 2times 15 times and how tall is this 18 okay so 2 * 15 * 18 we multiply it by two because there's front and back then we need to figure out the area of this shape right here and it looks like it's 45 by 18 so have 2 * 45 * 18 18 and to that I'm going to add the area of those triangles so have two triangles times times the base of my triangle looks like 15 and if you see back here the height of that is 12 so * 15 * 12 okay so when do some calculations here found out that the surface area is 540+ 1,620 + 180 which equals 2340 Square ft now the problem is the question doesn't ask us how many square feet is John going to paint the question is how many gall of paint will he need so we're going to take that total and we're going to divide it by 100 okay to determine the number of gallons now we know that when we divide by 100 that's the same thing as moving the decimal place over two times so he's going to need 23.4 gallons of paint probably it would be better to round up since you can't buy partial gallons of paint so we're going to say that John will need 24 gallons of paint okay all right let's do one more example this says find the volume and the surface area of the composite figure so let's talk about volume first it looks like have rectangular prism length time width time height and to that I'm going to add the volume of triangular prism which remember is the area of the base so times the area of the base which is triangle base times height times height of my prism so let's just plug in some numbers here okay so the volume of the rectangular prism is four * 2 * 3 and to that we're going to add 12 times the base of my triangle now if look at the base of my triangle can see that the base of my triangle here's the base and here's the height okay so have base of four and height of three so the base is four times height of three times the height of my prism now this right here is the height of my prism which is little bit different remember that this triangular prism is actually laying on its side so this really tells me how tall it is right here this tells me how tall it is and that measurement is two so we're going to multiply this by two okay so now let's do some calculations here all right 4 * 2 * 3 is 16 Sorry 4 * 2 * 3 is 24 8 * 3 is 24 plus 12 so the volume actually equals 36 cm and it's volume so 36 cm cubed all right now let's talk about surface area because that's going to be little bit different for the surface area have rectangular prism so my formula for surface area remember is 2 * the area of my base which is length width plus the perimeter of my base capital times the height of my prism okay and that would be for the rectangular prism and to that we're going to add the formula for for triangular surface area of triangular prism which is 2 * 12 base * height plus capital which is the area of the base times the height so those formulas we can plug in some numbers and get good answer however there's something that we need to think about surface area is covering an object and this piece right here this that I'm highlighting is the top of the rectangular prism and it is also side on the triangular prism we're not adding those so from this total from this total we are going to have to subtract this shape twice because I'm going to subtract it from the top of my rectangular prism and then the side of my triangular prism okay so the area of this shape is four this looks like 4X two if you see that is 4 * 2 so the area of that is going to be 8 so from this shape we're going to subtract 16 surface area like said in class can be little confusing especially when we talk about composite shapes because you have to think about where they match up all right so let's put some numbers in here have 2 * the length which is 4 * the width which is 2 plus the perimeter of my base well again let's look at the perimeter of the base here have four + 2 + 4 + 2 so that should give me 8 10 12 okay so the perimeter of my base is 12 and to that we're going to multiply the height of this prism which is three plus two times now let's calculate the surface area of our triangular prism so 2 * 12 times the base of my triangular prism we already discussed was four the height is three and to that we're going to add the perimeter now this is right triangle and know that three four and five make very nice Pythagorean triple we've talked about that several times okay if you didn't know that you would had to use the Pythagorean theorem okay so hopefully you remember that from class so perimeter is going to be 3 + 4 + 5 which is 12 times the height of my prism which is two and then from that whole thing we're going to subtract okay so my surface area area equal 2 * 4 * 2 so have 16 plus 12 * 3 which is 36+ 2 * 1 12 is just 1 4 * 3 is 12 + 12 * 2 is 24 - 16 all right so when figure all that out should get 72 cm and we're talking about surface area so that is squared okay all right so your homework is going to be to finish the problems that were on your worksheet from the 6.8 lesson 6.8 as well as the worksheet that's attached to your notes there are three questions attached to your notes good luck