Geo 10.3 Volume of Prisms and Cylinders

I was all right, welcome back, unit ten, we got volume of prisons and cylinders. So what is volume? It's the space that a figure occupies and is measured in cubic units. If we find area, we're finding what we can wrap around the outside. And that's two units times each other. So that's squared. Now we're going to find what fills these up. It's like how much milk fits in a milk carton. All right? So that would be three units multiplied together. And that's why it'll be in cubic units. So you can see how much space this occupies. Now here's a little clever little principle here. Cavaliers principle says, if two figures have the same height and the same cross sectional area at every level, then the volumes will be the same. So it doesn't matter if it's a right prism like this or a an oblique prism like this. If they are the same at the cross sections, which these are, these are called CDs for those of you who don't know, that's what we listen to before we had MP3s. All right. And iTunes and all that fun stuff. The heights are the same, the cross sections are the same, so it doesn't matter that it's the right prism or oblique prism. It'll give us the exact same volume, which is kind of nice. So let's take a look here. Volume of a prism is going to be times H big B is the area of the base. Remember that will change depending on the shape of the base times the height. Remember the height is the distance between two bases. So let's take a look at this one. The base here is a rectangle, so the forearm for a rectangle is length times width. Times height. Excuse me. So length times with 24 times 20, times our height is ten. Multiply all those together and I believe we get 4800 cubic centimeters. And centimeters and centimeters would be cubic centimeters. All right? Pretty easy. A lot easier, a lot less complicated than our surface area formula. Volume of a cylinder is the exact same formula. The only difference is now we know our base is a circle, so we pi R squared H all right, so let's find this one and look, this is not a right prism, it's oblique, but it doesn't matter. Pi times R squared R is our radius of our diameter two, our radius is one. Times the height of three. One squared is one. One times three is three. So this would be three pi meters cubed. And again, if we wanted to get an approximation of that, this is the exact answer, we'd have to plug it into our calculator and find out what three times pi came out to be. So let's try these. Let's do one at a time, so we don't lose focus here. All right, first one, we have a base that's a triangle and our height goes across is ten. So let's take a look at this triangle. I'm going to rearrange it a little bit. So this side is 8, this base is 8, and you can see this side is 8 as well. We need the height of this triangle. All right, well, let's see. This, if this cuts it down, this is an equilateral triangle. So when we cut this down, it divides us into two pieces. So we know now we now know this is 8, and this is four, and this is our height. So we can use the Pythagorean theorem H squared equals four squared plus H squared. 64 equals 16 plus H squared, 64 -16 equals H squared, 48 equals H squared, and take the square root of that. Let's see if we get the square root of 16 times the square root of three is four radical three. So now we can plug that in. So volume equals big B times H, what's the formula for the area of a triangle? It's one half base times height. Times our height now, let's remember this height refers to triangle. This heavy refers to the prism. So it's one half base, the base is 8, the height we found to be four radical three times the height of the prism, the segment that connects the two bases is ten. All right, well we multiply that out one half times 8 times four times ten. I believe we get a 160 radical three inches squared. All right? Let's take a look at the next one. Okay, this one's kind of different. Then we've done all year. This is a composite shape. In other words, there is a prism here that we can make, and then this is kind of a cylinder, but not the whole cylinder. It's half the cylinder. So if we want to find an volume, we're going to have to do base times height of the prism, and then we're going to have to add one half pi R squared H of the cylinder. So let's look at what we got here. So looking just at our prism here, let's just take this right here. That's 24 times 12. Times the height of the prism is ten. Plus one half times pi R squared. Now, what is R R? Well, this here is a diameter. And that was 12 if it's 12 years, 12 here. So the radius would be 6, 6 squared times H again, our H is ten, let's see what we get. 24 times 12 times ten. Plus one half pi times 6 squared times ten. So let's see 24 times 12 times ten. We get 2000 880. Half of 6 squared is 36, half a 6 squared times ten and we believe I believe that's a 180 pi. Inches cubed. Oh, this one was just cute. Sorry. Now this again here, right here, this is the exact answer. We could plug it into our calculator and get an approximation, which is probably a good idea, but right now, if I said, what is the exact answer, that's what you give me. If I ask for a approximation to the nearest tenth, you give me 3445 point 5 inches cubed. All right? Exact approximation. All right, mister bras almost made the Olympics as a swimmer, swimmer actually lost the folks in his first race at the trials. I mean, among other people, he lost the Phelps and others. Just go ahead and ask him about it. It's pretty amazing. He's a really great swimmer at the university of Miami. You can see that's the Miami colors right there. All right. He's trying to figure out the volume of the pool during the race or so he claims. So let's see, the pool, a pool is a prism, right? So we can draw our prison. Wow. All right. So how much water does the pool hold? So it's 50 meters long. It's 25 meters wide, and only two meters deep. So volume equals big B times H so it's going to be 50 times 25 times two and that gives us 2500 cubic meters of water. In other words, a lot of water, all right? So I want you to pause the video, try these two on your own, and then we'll see how you did. All right, welcome back. So here we go. We want to find a volume and that is big B times H let's see, what is our big B is a triangle, the base of the triangle, so the volume or the area of a triangle is one half base times height times height. All right, so let's take a look at this triangle. All right, here's a triangle. Well, this is the height, so one half times the height is 7. That means this is the base, which is the same as this one, so it's 11. Times the height of the prism, the distance between the two bases is 6. So one half times 11 times 7 times 6, and I believe you get 231 cubic feet. All right? Let's find this one. Volume equals pi R squared H R squared R is going to be three because the diameter is 6. Three squared times H out with no H H is three. Three squared is 9, 9 times three is 27. Miles, cubed. A cylinder you'll never see in your lifetime. 27 miles cubed. All right, there you have it. Good luck on a 10.3 master check. As always, try to learn the material before you take the master check people. Why retake it if you don't have to. You know we're not going to let you move on, so you might as well learn it the first time. First of all, see you next time. Prism rise. Yeah. Whoo. Let's go. Treatments like this we have tonight 'cause they shouldn't even die this time but the height has to be right and go to the constitution. That's a pretty good time but the height of the feeling and we'll do the perfect choice. And the dream is a problem. Looking forward to values. Crab section has yeah and every year 16. Here it goes volume is 8 times 8 5 1680. Yo. Two dimensions are enough for me. I got these freedom mentioned and there's a rock. I'm happy this prison rock it's got its own formula now link times red time type but only if it's rectangular. Just security area big time but the highest it feels right to the constitution all right prison rocking the hands tonight let's take a time look at how that could be right and we do the prosecution value every day and figure in. Step up quick. What's the volume of this freak type of prison how do you check? It's tough. Triangular. It's 24 harder. It's tense. It's two 40. One to the side. One two four four three four three four two and we got the phone for you for you for you prison I think in the last 6 years. But the highest feeling right they were through the constitution for four months wow oh. Oh oh oh. Girl you got the body every day I'm fig figure in. Yeah.