Math 10 C Surface Area and Volume of Composite Objects

hello and welcome to video on the surface area and volume of composite objects composite objects are objects like not regular shapes but shapes that are familiar to us that have been added to other shapes so just as by way of an example when we look at the shape on the screen here we see we see pyramid that seems to be it's it's pyramid here that's stacked on top of rectangular prism or square base prism so and it's saying to find the surface area of the composit object well the surface area what we want to do is we want to try and identify all of the surfaces that are showing or that are present in our object now if this was just regular pyramid and sorry and regular prism when we take one object and we stack it on top of the other we end up missing or we end up having two of our faces two of the surfaces disappear because of the stacking so the bottom of the pyramid and the top of the prism are no longer showing they're no longer like present for us to either paint or cover or whatever we'll for whatever reason why we're calculating the surface area so we can't take those sides into consideration well what else do we have what other sides do we have well we've got we've still got our four triangles from our pyramid and we've got the sides of the prism and then we also have the bottom of the prism as well so the sides of the prism hopefully you'll notice that this is square based prism so the bottom is length times length and the sides because they share the same height it's 12 centimeters high and they share the same base all four sides around this object all four sides here at the back and on the side at the front they're all the same so if we want to add up the surface area of this object it's the four areas of the triangle plus it's the four lateral sides so the four length times height plus it's the one bottom and that's length times length or we could just write that as length squared so that's the idea it's the four sides of the triangle it's the four sides the lateral sides of this prism in the bottom let's go ahead and calculate that four and its base times height divided by two so when we look at the base of this triangle it's six centimeters wide so six centimeters the height to the triangle is eight centimeters divided by two plus four length times height so six times 12 plus just the bottom is six squared when calculate this end up getting 420 square centimeters so centimeters times centimeters is centimeters squared same thing here same thing here so we end up getting 420 square centimeters perfect let's try another example for volume volumes little bit easier than surface area don't really care that this circle on the bottom of the cone and the circle on top of the cylinder are not being presented because that doesn't affect the volume all we have here the volume of this object is just the volume of the cylinder plus the volume of the cone so it's just whatever the cylinder is and the cone together so the volume of the cylinder is going to be - sorry it's PI squared and then the volume of the cone is gonna be PI squared divided by 3 now we probably should make little bit of we should discriminate between the height of the cylinder and the height of the cone because they are two different things so we'll say height of cylinder and height of cone here just to be sure so the volume is PI the radius is the same either way it's seven squared and the height of the cylinder is 15 plus PI sorry is seven squared and the height of the cone is 25 divided by three when calculate this when put this into my calculator end up getting three thousand five hundred and ninety one point eight eight seven six seven six zero one sorry that's one too many sevens six zero one blah-blah-blah-blah-blah and this is centimeters cubed and it says we want to round to the nearest hundredth okay we can round to the nearest hundredth this is our hundredth right here and since the next digit is five or higher we're gonna have to round this up to nine three thousand five hundred ninety one point eight nine cubic centimeters let's try another example says determine the surface area and the volume so we will do the surface area and blue and the volume and red determine the surface area and volume of the composit just below round the values you can't see it it says to the nearest tenth no problem so we round to the values to the nearest tenth is what is saying so let's try the soap the surface area of this object is going to be the outside the lateral area of the hemisphere plus the lateral area of the cone so those are the two sides being presented to me the surface area is the area of the hemisphere being presented plus the area of the cone that's being presented so hemisphere is just half of sphere so instead of four PI squared it's going to be two PI squared and then the area the cone where we're wondering okay is it the PI squared or is it the PI that we're thinking about and it's it's definitely not the PI squared so that is not the section because the PI squared would be the circle on the top of the cone but we don't care about that because it's being hidden by the hemisphere so the surface area is 2 PI squared okay 2 PI is 8 squared and then the lateral area the so that's the area all the way around this cone is PI which is 8 and the slant height the slant height here we're telling us 17 centimeters so when plug that into my calculator end up getting 829 point 3 8:04 blah-blah-blah-blah-blah centimeters squared so if want to round to the nearest tenth that three is gonna the eight sorry is gonna make the three round up to four so we end up with eight hundred and twenty nine point four centimeters squared now let's try for the volume the volume of our kind of looks like an ice-cream cone guess is the volume of the cone plus the volume of hemisphere so the volume is gonna be the volume of the cone plus the volume of hemisphere the volume of cone is if you recall back to one of our last videos it's PI squared divided by three and then the volume of the hemisphere is half of the regular volume of sphere so that's 2/3 PI cubed 2/3 PI cubed so we can try this so hopefully that won't be that big of deal volume PI is 8 so 8 squared times the height of the cone that's 15 centimeters divided by 3 and then 2/3 pi and then the radius square cube sorry when punch this into my calculator end up getting volume of 1139 point 3 5 0 9 buh-buh-buh-buh-buh centimeters cubed again we want to round to the nearest tenth so the 5 is going to make it round up so our volume will be 1139 0.4 cubic centimeters and then our last example it says determine the surface area okay we'll do surface area and blue again and the volume ok we'll do volume in red and it says round our answers to the nearest tenth again no problem so the surface area well what surfaces do have being presented it looks like we have circle on the bottom that's going to be PI squared looks like we have the lateral area of the of cylinder no problem and then we've it's almost like we have scoop cut out in the shape of hemisphere and it doesn't matter if it's cut out because that surface of the hemisphere is still being presented to us so the surface area of this object is going to be the hemisphere well that's half of sphere so that's two PI squared plus we have the sphere or the circle on the bottom so that's PI squared plus we have the lateral surface area of cylinder so that's that all the way around it and that is that is 2 PI ok well 2 PI squared plus PI squared is really just 3 PI squared so that'll give us little bit of pre from our calculation 2 pi RH so the surface area 3 PI we know is 8 squared and then 2 PI is 8 again the height is 20 so when put this into my calculator the surface area end up getting and we want to round it to the nearest tenth so one six zero eight point four nine five blah-blah-blah-blah-blah the tenth is our four so our four is gonna round up to five so we'll just change that to five and this is centimeters squared perfect let's try our volume now our volume is is the volume of the whole cylinder but it looks like we've scooped out this hemisphere here so it's like we're gonna take the volume of the cylinder the volume total is the volume of the cylinder and then we've scooped out the volume of the hemisphere so the volume of cylinder is 2 sorry as PI squared and we're gonna take away the volume of hemisphere that's 2/3 PI squared so we just have to plug in these values PI is 8 squared is 20 minus 2/3 PI is 8 again PI cubed sorry it should be cubed so when plug this into my calculator end up getting two thousand nine hundred and forty eight point nine zero eight Pablo blah blah blah we're number one around to the nearest tenth our nearest tenth is this nine here so we can just keep it as nine and this is centimeters cubed okay well hopefully that helped