Common Core Geometry Unit 10 Lesson 10 Spheres

Right a sphere really truly is the three dimensional equivalent of a circle. Because it's the collection of all points in three dimensional space that are equally distant from a fixed point, right? That's exactly the same definition as a circle. The only difference is that a circle is in two dimensions and a sphere is in three dimensions. But every point on the sphere, any point on the sphere is the same distance away from its center as every other point. Now, the volume formula for a sphere only depends and this should make sense on that radius. That is really the only thing that determines what a sphere is, is the size of its radius or diameter, depending on how you want to think about it. And the volume formula for a sphere is four thirds pi R cubed. Now in the last lesson, we looked at the volume of a pyramid and a cone, and we saw that they were one third the base area times the height. We're not going to even try to justify the four thirds in this particular lesson. 

We're going to state the formula as such. And then we're going to work with it. It truly, in my personal opinion, takes calculus to understand why it's four thirds and not four fifths or 7 halves or something like that. But at least there's a third in it. And that's very similar to pyramids and cones. Anyway, let's jump right in and do some application problems with this. Exercise number one, straightforward. Let's find the volume of a sphere whose radius is 6 inches, first in terms of pi, and then rounded to the nearest cubic inch. Let's do this together. All right? The volume formula for a sphere is four thirds times pi times R cubed. So in this case, we have four thirds times pi times 6 to the third, all right? And that's going to be four thirds times pi times 216. Now, I want to get this in simplest terms of pi. I hate to say this primarily for one reason, which is that it could easily show up as a multiple choice question. Now, I realize that it's four thirds times pi times 216. But what I'm going to do is I'm going to multiply the four thirds in the 216. And I'm going to do that on my calculator. That's okay. I'm just going to do four thirds. Times 216. And it's 288. That's it. So now, as I come back into here, I can now say that the volume. Is 288 pi. And specifically cubic inches. Okay. 

Now, that's good. You want to be able to do it, because quite frankly, it could show up in terms of multiple choice and things like that. But in every practical scenario, you really want your volume in terms of something that you can visualize and something you can use, which means that we got to get rid of that pie. We got to have it to the nearest cubic inch or something like that. Now, if that's the case, of course, all we really have to do at this point is due to a 188. Times pi. And there's our actual volume. 900 and to the nearest cubic inch 905 cubic inches. 905 cubic inches. Thankfully, most of the time, this is the way that we're going to want that volume. A volume that we can use that we can actually do something with, as opposed to left in terms of pi, which is good, it's exact, but you can't do much with it. Anyway, so take a look, and then we'll move on to the next problem. All right, well, that's a real kind of plug and chug problem. Let's see if we can get something a little bit more interesting. All right, exercise number two. A sphere has a great circle. We'll talk about that in just a minute. 

A great circle whose circumference is 95 centimeters, which of the following is closest to the volume of the sphere in cubic centimeters. All right, so basically it works like this. If you take a sphere and you slice it, you do a cross section of it in any direction in any place you get a circle. It makes sense, right? But every sphere has these things called great circles. Okay? And that's when you slice the thing right through its center. Think about the earth for a second. The equator is a great circle. Any line of longitude, not latitude, but lines of longitude. So like the circle that would connect the North Pole to the South Pole and then go back around. That would be a great circle. Okay? So what we're told in this problem is that this sphere has a great circle whose circumference is 95 centimeters and we want to try to figure out based on that. What the volume of this particular sphere is. Or at least which one of these is closest. Pause the video now and see if you can figure that out. All right. Well, ultimately speaking, it's always important to know what you need. 

So since I'm trying to figure out the volume of the sphere, I'm going to write down its equation. Four thirds pi R cubed. It means I need to know what the radius of this sphere is. But the radius of the sphere will also be the radius of this great circle. And what do I know about this circle? What I know about it is that its circumference is 95 centimeters, right? And its radius will be the same as the radius of my sphere. Now, how can I figure out what that radius is? Well, I know that the circumference formula for a circle is pi times the diameter, or two pi times the radius. That's the circumference formula for a circle. So if I set that equal to 95 centimeters, I can then divide both sides by two pi. And I can figure out the radius of my great circle, which is also the radius of my sphere. Now, do we do that now? Get some long destiny mess. Messy decimal, or whatever. Or do we kind of leave it like this? I think I'm going to actually figure out what this is for right now. 95 divided by two pi. Let's take a look. Clear this out, 95 divided by two pi, notice how I'm putting that two pi in parentheses that guarantees that the 95 is divided by both the two and the pi. I hit enter. And there's my radius. 15.1197. Et cetera. It's going to get a little messy now. Let me just check on that 15.1197. All right. 

Now one thing I could do with this, though, is I could store it somewhere. In fact, I can use the store button on my calculator that will take the answer, and maybe I would even store it. In the letter R, R for radius. And now I've got that radius stored in the variable R and I can use that later on. How do I want to use it? Well, now, whoops. Now if I can get back into my program, right? The volume of my sphere now will be four thirds times pi times that 15 .1197, et cetera. Cubed. Make sure that looks like the decimal. Right? So let's go back to our calculator and do that. The great thing is I've stored this radius in R so if I now go to my calculator, I can do four, thirds, times pi, times, R to the third. Four thirds pi R cubed. Enter. And there it is. 14,478. 14,478. Which one is closest? And the closest one is 14,000. 500. Pretty close. There we go. Let me have you take a look. And then we'll move on to another problem. All right, let's do it. What do we have next? Oh, I like it. We put a sphere in a cube. Let's take a look at what the problem asks us to do. The largest sphere possible is placed in a cube that has a side length that measure 8 centimeters each. What percent? What percent of the cube's volume does the sphere occupy? Round to the nearest percent. All right, so is this sphere taking up 50% of the volume of the cube? Is it taking up less, 25%? Is it taking up more 75%? Play around with this and see what you find. 

All right? And then we'll walk through it. All right. Well, I hope you remember how to do percent. It's always part divided by total times a hundred. The total here is the volume of the cube. So let's figure that out. The volume of the cube is going to be 8 times 8 times 8. No great surprised. It's 8 cubed. All right? Let's go grab that really quick. Let's do 8 cubed. And 512. All right. 512 cubic centimeters. Now let's figure out the volume of that sphere. Now obviously the volume of the sphere. Is going to be four thirds, pi R cubed. What we have to know is what the radius is, yet again. Hopefully that was obvious. The radius of the largest sphere that's going to sit inside of this cube is going to be half of the length of the cube's side. Right and think about that. That's because one part of the sphere touches that side. The other part touches that side so the diameter of the sphere will be 8 centimeters. The radius of the sphere will be four. All right? So that will be four thirds pi times four cubed. All right? Now this is going to be quite messy. All right, but let's take a look on our calculator. All right, four thirds times pi times four to the third. Let's do it. 268.083. Two 68.083, let's say. 

Those are also cubic centimeters. So I want to know what percent this is of this. Now again, I think that most students can handle this, but just think about it. It's almost like you were taking a test that had 512 points on it. You got 268.083 points. What grade did you get? Well, it should be as simple. As basically just saying the percent is going to be two 68.083 divided by 500 and 12 times 100. And my red. There we go. All right, so let's do it. And quite frankly, it's going to be quite easy now. Because I've got that two 68 sitting there. Now I just have to divide it. By my 5 12, many students will be able to look at this right now and say, well, that's 52.35%. But if for some reason, that's hard for you to see, just multiply by a hundred. And there's our percent. 52.35, et cetera. What did we want that to? To the nearest percent, so that's going to be 52%. Strangely enough, just barely over half, right? Just barely over half. To me, when I look at the picture, and I think about it, it feels like it's going to be more than that. I feel like it should be more like 70% of the volume or something like that. But it's just barely over half of it that's taken up by the sphere. All right. Let's keep doing it. Actually, let me pause for a minute. Take a look at any of this. And then we'll move on. 

Okay, let's do it. Number 6, now let's start putting some shapes together. A grain silo stores corn can pack at a density of 16 pounds per cubic foot. The silo is shaped like a cylinder, along with a hemisphere, a half of a sphere. Of the same radius on top of the cylinder is shown. Determine the volume of the silo to the nearest cubic foot. All right, well, you know how to calculate the volume of a cylinder. You know how to calculate the volume of a sphere, so I'm hoping that the volume of a half of a sphere isn't too hard. You just have to put the two together. Spend a few minutes doing that. All right, let's go through it. So basically, just like this, let's go for it. The volume of the cylinder is going to be the base area times the height. The base area is that circle pi times 5 squared times its height, which is 15. All right? And again, we could kind of work this out. We could leave it that way. It's kind of up to you. So or we could leave it even in terms of pi. That might be kind of nice. If we want to leave it in terms of pi, you would simply let's clear this out. You'd have the 5 squared times 15, so it would be 375 pi. Or you could multiply it by pi now. I'm going to leave it as 375 pi. So that's the volume, just contained within the cylinder. 

Now I have to have the volume contained within the hemisphere. So the volume of the Hemi will just do it that way, is going to be one half times four thirds pi R cubed. The one half is simply sitting there because it's half of a sphere, right? Now you could, of course, play around with that half of four thirds is two thirds. If you wanted to do that, or you could just leave it as one half times four thirds pi. And then the radius is the same as the radius of this 5. All right. Oh boy, that's kind of getting ugly. But let's do this. The volume overall let's call it the volume total. Will be the volume of the cylinder. Plus the volume of the hemisphere. Okay? So we still need to have that volume of the hemisphere. I say that we kind of figure them out now. So let's do that one. One half, four thirds pi R cubed. So let's do this. .5 times four thirds times pi times 5 to the third. Again, just keep in mind. That's one half times four thirds times pi R cubed. Our radius was 5 feet. So 5 cubed enter. All right? Added to that is going to be the volume of that cylinder. Which was 375 pi. So plus three 75 times pi. Gives us that total volume. 1439.89, et cetera. You see if that's right, yeah. So it's 1440. 1000 440 cubic feet. Always remember your units. All right. Now, let her be. How many tons, let's go back to here. 

How many tons of corn can the silo hold if there are 2000 pounds per ton? Round to the nearest tenth of a ton. All right, well, wait a second. So most of the time when you're talking about large quantities of animal feed or corn or wheat or whatever, you talk about it in terms of tonnage. There are 2000 pounds per ton. But we don't know how many pounds there are yet, except we're told up here that the corn will pack at a density of 16 pounds per cubic foot. And we know we have 1440 cubic foot feet. So we can figure out the amount of weight we have. By doing that 1440 cubic feet with a little red thrown in there. Cubic feet times 16 pounds per cubic feet. Now we've got to go back over to the calculator to get that. Again, remember that 16 pounds per cubic foot is given to us up here. It's called the density. So we're going to go back to here. 1440. Times 16, and that gives us 23,040 pounds. All right? 23,040 pounds. But we want to know how many tons of corn feed. We know that there are 2000 pounds per ton. So now we can just say that the tons will equal 23,040 divided by 2000, let's go back over to here. Divided by two thousand equals 11.52 tons. I think it said round to the nearest tenth of a ton, so 11.5. All right. 

Let me step back, maybe scroll this up a little bit so that you can see them. Write down anything you need to. All right. One more problem. Here we go. All right, how could we possibly talk about cones and spheres without talking about an ice cream cone? Let's take a look at what this problem asks us to do. Exercise number 5 Nico got the jumbo ice cream cone at the duchess county fair. The amount of ice cream can be modeled using a cone whose height is 12 centimeters and a radius of four centimeters. A hemisphere and then a full sphere of ice cream, the same radius were loaded on top of the cone as shown. If there are 237 cubic centimeters in a cup and 320 calories per cup of ice cream, then how many calories is this treat? If the cookie itself has 85 calories. Oh boy, how many calories does it have? All right, so this is a pretty involved problem. What I'd like you to do is pause the video now and see if you can figure it out. Take a little bit of time on this. All right, let's go through it. Let me take a look, by the way, the final answer is 899 calories, almost 900 calories. And this is a pretty realistic problem, by the way, so watch those treats at the duchess county fair. Have only one of these things. All right, but here we go. Let's see where we come up with about 900 calories in this particular treat. All right, so we've got three different volumes of ice cream. We've got the volume of ice cream in the cone. 

The volume of ice cream in this hemisphere, and then the volume of ice cream in this sphere. And we have to figure out what each one of them is. All right, so here we go. Let's start by doing the cone. Okay, the volume of the cone is going to be one third times its base area times its height. Simple enough. No, no pie yet involved, but while I have a little bit, so we've got one third times the base area. Now the base area of that cone is going to be the area of this circle. Okay, so that's one third pi times four squared times its height, which is 12. All right. One third pi R squared times H, the volume of a cone. Let's actually go through that really quickly. Let's get that answer. So clear this out. One third times pi times four squared times 12. All right, times four squared times 12. All right, what do we have? We've got a total volume of ice cream just in the cone of 201.06. We'll just take two decimal places. 201.06 cubic centimeters. All right, so that's the volume of the cone. Now we've got to have the volume of ice cream that's in the hemisphere and the volume of ice cream that's in the sphere. So let's do the volume and the hemisphere. That's going to be a little bit of a challenge because that's going to be one half times four thirds times pi R cubed. So that's going to be one half times four thirds times pi times four cubed. So let's take a look at what that is. 

One half times four thirds times pi R cubed. All right, .5 times for the divided by three. Times lots of messy calculations here. Times four to the third. All right? Again, one half because it's a hemisphere four thirds because it's a sphere pi, and then R cubed. All right, what do we get there? A 134.04 cubic centimeters. Let's note that one 34. One 34.04 centimeters cubed. And finally, and then this one's going to be fairly easy is just the volume of the ice cream in that sphere. Which right is just going to be four thirds pi R cubed. You seem to have a little bit of a buzzing going on in here, but hopefully hopefully it will be gone soon enough. Or not. So four thirds pi times four cubed. All right, four thirds times pi times four to the third. And that gives us two 68.08. Now, by the way, this volume is no great surprise exactly twice this because the two hemispheres now, whether this is completely realistic or not as questionable. But the two, the hemisphere and the sphere are exactly the same. This is just half of that. So two 68.08. Two 68.08. Cubic centimeters. Wow. 

So now, of course, what we need to do is we need to add all of those up. And that's not too bad. In fact, I kind of did it all. Just making sure I've got 201, one 34, two 68. It's all good. I'll save you going over there, but the volume total volume of ice cream. The total. Is going to be 603.186. Or let's go 603 .19. We'll just go with that. Cubic centimeters. All right, so when we add all of these together, let me just double check that. Yeah, we'll go 603.19 cubic centimeters. Now, that's great. But ultimately, I'm trying to figure out what the volume of the or sorry, how many calories are in the ice cream. I was told I have 300 calories per cup of ice cream, but I need to know how many cups I'm eating, all right? For that, we were told that each cup contains 237 cubic centimeters. So we just need to divide this by 237 cubic centimeters, and we will find out, all right. So what was that? It was 6 O 3.19, 6 O three, .19. Divided by 237. Equals 2.54 cups of ice cream. Now, I think we've all seen a cup measuring cup. You know, a cup of ice cream doesn't seem like that much to me. In fact, I can routinely eat maybe more than a cup of Ben and Jerry's in a given night. But what we've got is we got 2.54 2.545 cups. 2.545 cups. So how many calories of ice cream do we have? Well, we've got 320 calories per cup. So if I've got the 2.54 cups. Right? We've got 2.545 cups. 

Times 320 calories per cup. This is not light ice cream. And that will equal. Let's find out. Times 320. Wow, a whopping 814 calories in the ice cream alone. So that's 814 calories. It seems a little bit silly perhaps, but now we're going to add in the final amount. Which is the fact that the cone itself made out of a cookie, a delicious delicious cookie, is 85 calories. So we add on an additional 85 calories, and we get 899 calories. A little bit of a long problem, but we got it. Almost 900 calories in that particular dairy treat. If you live in upstate New York and you can get to the duchess county fair, you should. It's a wonderful fare. One of the best ones around. But this problem is pretty involved, right? The volume of the cone, the volume of the hemisphere, the volume of the sphere, add it all up to the other. We get a volume in cubic centimeters. But we normally think about things like ice cream or milk or other things like that. 

Their volumes in terms of cups, we have that conversion factor, so we are able to change it into cups, changing the cups into calories, adding on the cone, and that's it. All right, take a look, and then we'll wrap up. All right, let's do it. So what did we see today? What we really saw was one more formula. The formula that tells us how to calculate the volume of a sphere. Four thirds times pi R cubed. It's a messy formula. It's got a fraction in it four thirds. It's got pi in it. It's got a cubing in it. It can get messy. And I apologize if that last problem got a little bit messy with all the different volumes and things like that. But at the end of the day, it's still just keeping track of all your numbers being careful about your work and organized about your work and you can pull these problems off. All right? We've got one more lesson in the curriculum and we'll get to that next time when we look at the volume of a truncated cone, truncated cone, we all know about those. Anyway, for now, I'd like to thank you for joining me for another common core geometry lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. Keep solving problems. And keep eating ice cream.