Finding the surface area and the volume of a sphere

are so let's first jump in and just look at our note taking guide and fill in some of our information here so what is a sphere a sphere is the set of all points in space equidistant from a point okay it's very similar to a circle when you're talking about a circle you're talking about all points equal distance from a point but it's in a flat plane when you actually make that into uh three dimensions we still have that Central Point in the middle but I always swing a lanyard in my classroom to kind of demonstrate it okay my hand is the center of the sphere and they start swinging the lanyard in different directions and what it creates is not just a circle but it creates a sphere too because it creates all the points that are left and right up down and in Corners so that's how I demonstrate my sphere in my classroom okay what is the radius of a sphere well that's pretty easy it is from the center the segment from the center to any point on the sphere and we know that those are equidistant because from the center to any point on the sphere it's just like the radius of a circle but we have to think about it in three dimensions okay great circle great circle is not a circle that's drawn really well it is actually the largest Circle that can be drawn in a sphere and so I've actually got a little picture to kind of show you what I mean okay if you have a sphere and you start slicing it okay the great sphere is not this one up here or the great circle is not this one up here that's not the largest Circle that can be drawn it would be like the equator in the center of the sphere it is the largest Circle that can be drawn on inside the sphere so it is the largest Circle that can be drawn in a sphere so you're slicing a sphere through the equator and so an equator is really the best example of what the of where the great circle would fall so if you have a sphere like on the earth the equator is right there at the the very center point and it is the largest right the largest Circle that can be drawn if we slice a sphere okay hemisphere what is a hemisphere well you've heard of it in science a hemisphere is just what half of a sphere Hemi means half just like semi so it's half of a sphere so if we take a sphere and we cut it in half we have a hemisphere okay so like an eagloo is a good example I like to draw my little eagloo cuz that makes sense to me okay so let's look at a couple formulas that we're going to be using using okay first of all is surface area surface area is Just 4 pi r s and if you look p piun r s is just the area of a circle so all we do is take it multiply it by four to get the surface area so it's pretty easy okay now for the volume your volume is threedimensional so you're going to actually Cube your X your radius so be very careful biggest mistake kids forget to cube the radius invol volum instead of square it so it's 4/3 piun R cubed and that's the volume of a sphere okay so let's look at a couple examples here says find the surface area of a sphere that has a radius of nine give both the exact answer and the rounded decimal answer so an exact answer always has a pi in it cuz that's the exact value it's not estimated or rounded remember when we actually multiply Pi out it's it's an estimated answer okay so let's find the surface area so we're going to use our surface area formula so let's write s = a um sorry S = 4 pi r 2 not a pi r 2 okay so we're just going to take 4 * pi and then our radius is 9 so we're just going to square 9 so 9 squar is 81 so we need to take 4 * 81 so let's not make any silly mistakes let's go over here and do four * 81 and I get 324 so we're going to get 300 and 24 pi as our exact answer okay and that's an in squared because it is area now we need to do the estimated or the approximate answer so all we're going to do is take our 324 and we're going to do multiply you you can just leave it in there and be times pi which is second function and carat and we get 17.87% so that's an approximate okay so there are both answers and you need to be able to identify them in both forms if you get to a standardized test and've got it in terms of Pi know what it should look like and then you need to know what it looks like multiplied out also okay now let's do a volume problem number two says find the volume of a sphere that has a diameter of 10 Okay so we've got this time a diameter and not a radius and our formula for volume let's slide this down for just a second our formula for volume involves the radius so we're going to have to figure that out first so if the diameter is 10 our radius is what five okay so now we've identified our variable we write our formula down 4/3 pi R cubed and remember I told you a lot of times students forget to cube that so I'm going to have 4/3 * < * 5 cubed okay what's 5 cubed please don't say 15 it's 5 * 5 * 5 which is 125 so 4/3 * 125 Pi okay when we have a whole number we stick a one under it so 4 * 125 is going to give us 500 yep so 500 over 3 so our exact answer is going to read sorry 5003 pi and that's cm squared that's 5003 Pi cm squared to find an approximate we'll take 500 / 3 time our PI button and we're going to estimate that to 523 what's that 60 yeah cuz 5 98 would round to 60 so the approximate value is 5 23.6 CM whoops cubed cubed not squared cubed okay so there is how we use our formula just going forward I always think of a forward and backward so it's easy we just plug in and do the formula forward