okay this is our next section surface area and volume of spheres so we're going to go straight to the formula the surface area formula for a sphere is 4 PI R squared 4 PI R squared and I like to relate it to a baseball because if you're familiar with a baseball the way it's stitched and you unravel it it looks like there could be four circles okay which if you see in the formula PI R squared represents the area of a circle and then you have four circles four areas of circles so something to think about I know you've got that stuff on the inside but if you were to kind of level it all out and then the volume formula for a sphere is 4/3 pi r RQ Easy's we're just going to go ahead and do two examples using these formulas so the first one is just a regular sphere the radius is 27 so we're gonna plug that in and just work each formula how pretty it's pretty easy all right so let's do surface area first so surface area equals 4 PI R squared 4 PI 27 squared so the exact surface area is 2916 pi and this is millimeters squared okay pretty easy the superball you know volume is 4/3 PI R cubed so I'm going to put seven in there is a really big number I just double-check that so nineteen thousand six hundred and eighty three at then times four-thirds so I'm gonna just do that in my calculator which comes out to twenty-six thousand two hundred and forty four five so this is millimeters cubed okay and that's the volume one thing that I do want to and it's what is the radius and you had to solve for that all you need to do is set up the formula 4/3 PI R cube and set it equal to whatever and then solve for it so for it are using your algebraic steps first thing that I would do with tax class of pies and I get rid of the 4/3 so the way you get rid of that is multiplied by 3/4 and so then we have R cubed equals 50 500 times 3/4 4125 and here's what some kids get confused they don't know how to get rid of the cute it's very similar to the way you would get rid of the squared except it's called a to Brooke but that right there is in your calculator and so whatever you do to one side you got to do to the other and that'll get rid of that and then it'll get R by itself so the way you cube root if you turn your iPhone sideways you can see a cube root on there so this is roughly about sixteen point zero three eight okay so that's solving backwards for our if you were ever given the volume all right let's try this sphere over here it's actually not a full sphere that's a portion of a sphere it's like you got a little portion cut out so when we do the surface area first over here a little bit so when we do the surface area we don't want to quite do 4 PI R squared okay because you have to consider a lot of different things I always like to relate it to painting you have to paint this inside you have to paint this part right here which right here this is a quarter of a circle and that's a quarter of a circle cuz up that's the center of the sphere and if we have a sector right here okay so let's take care of this bottom part this bottom part is easy to do that's two PI R squared I took half of 4 PI R squared and then I need to add this curved part up here that curved part is a fraction of 2 PI R squared so we're going to add a fraction of 2 PI R squared and the fraction is whatever the degrees is all the way around like that and if this is 30 degrees the remaining part must be 300 at 30 so 330 over 360 times 2 PI R squared so that's those two parts now hang this up a little bit get rid of that so now we just need this inside part that I'm now going to highlight in red so this right here we just said it's half a circle of a circle here's another quarter of a circle so that's half a circle we're going to add half of a circle which is half of PI R squared and then the last thing that we need to add is this little sector right here that i just put in green which that is is acceptor which is 30 degrees 360 times PI R squared so that's the entire formula for all the sections that we need to find okay so let's start off with the easier one first the two PI R squared and our it's four right there so this is the first part so 2 pi 4 squared plus I'm going to reduce this fraction up here it's 11 over 12 11 over 12 times pi 2 times 2 pi 4 squared so let's do that and we'll do the red and the green in a second so so this is 32 pi plus 16 4 squared and then times 2 is 32 and then 32 times 11 divided by 12 is 29 and 1/3 the 29 in 1/3 pi ok so I'm going to switch my color to red because this is the next one so we're going to add 1/2 PI 4 squared and then we're going to add let's reduce that 112 alright so 12 times pi 4 squared all right so I have so I have 32 pi I'm just going to rewrite it one more time 32 PI plus 29 and 1/3 PI plus 16 and then 1/2 that's 8 pi and then this last green one 16 and then 1/12 16 divided by 12 so 1 and 1/3 PI so now all of these are like terms and that's our surface area because we did every little piece that we needed to so to speak paint okay so now I'm going to add 32 plus 8 is 40 40 plus 29 plus 1 at 70 and then 1/3 plus 1/3 is 2/3 so your surface area for this sphere that has the portion cut out is 70 and 2/3 PI square feet okay and I know that seems a bit complicated but you have to take each little piece into consideration because that's what surface area is alright so now to try volume for this figure the volume you kind of want to do each hemisphere one like one at a time okay and or you can do you can do the whole thing if you want as well but I like to consider it all together at one I mean uh each little half one you know at a time so for volume or volume its 4/3 but I'm going to do each hemisphere at a time so this piece down here it's 1/2 4/3 PI 4 cubed and then at the top we're going to do the other half 4/3 PI 4 cubed except I got to take a portion of it ok and it's not 30 degrees it's 330 degrees out of 360 ok so you want to do that portion right there of that top hemisphere so what does this reduce to so this reduces to 2/3 so we have 2/3 times what's 1464 hi Plus remember earlier we did we already did 330 out of 360 we reduced that so that reduces to 11 12 times 2/3 because that's what reduces there times 64 times pi so now I just need to reduce those right there so what's 2/3 of 64 42 hi plus when I'm with this multiply these in my calculator 11 12 times 2 34 God so this is thirty nine point one one one repeating what fraction is that I like to turn things into fractions let's see if I can figure out the fraction oh it's 14:08 over thirty-six so this is really 39 and 1/9 just to be exact mm-hmm and then of course we have pi so the way we write our final answer is that's roughly eighty one point seven seven eight hi okay if you want to turn it into a fraction you can turn it I'm just going to leave it as that all right and this is the volume so this is in cubic feet