Understanding Surface Area and Volume

In this video, you're going to learn how to find the surface area and the volume of three-dimensional figures. We're going to talk about cones, we're going to talk about pyramids, we're going to talk about prisms and cylinders and spheres. So let's get into this video and we're going to go through the formulas. I'm going to show you an easy way to remember these formulas and how to group them together so that you don't forget them and you'll be able to solve these problems easily. So let's get into the first group of problems. The first group, we're talking about prisms. And basically what a prism is, Prism is a figure, a three-dimensional figure, where the two bases are parallel, okay, and they're congruent. So that means they're the same size and shape, and they're parallel, they don't cross, okay, if they were to continue to go, and they're separated by the height, okay. Now I group cylinders, okay, with these prisms because a cylinder is really like a circular prism. You can see the two bases are circles, and they're separated by that height. So when you think about cylinders and prisms, think of them all as one group. So we're going to do the surface area and the volume. So first of all, let's do the volume of this figure. Now when you do the volume, the formula, the general formula, is very easy to remember. It's the area of the base times the height. So that's why I did a capital B. So the area of the base means you find the area of the bottom, that's what it sits on, and you multiply by the height. So essentially what this is like, it's like taking the area of this square in this case, okay, which is 16 inches squared. And it's like taking those squares, like a stack of sticky notes, and you stack them up, okay, 10 inches high. So you take the area of the base, which is 16, times the height, which is 10, which is going to give us 160 inches cubed. Now, the reason it's cubed is because, you know, we're filling these up with, like, little ice cubes you could think of, okay, something like that, right? So little one-by-one-by-one cubes, and that's how many are going to fill up the inside of this box, right? Now let's talk about the surface area for a moment. So the surface area is like if you were going to take a paintbrush and you were going to paint all the sides, you know, of this figure. So when we think of the surface area, this is the formula that you want to memorize. It's 2B, okay, the capital B means the area of the base, two of them because you have a top and a bottom, plus the perimeter of the base times the height. Now let me see if I can show you a three-dimensional figure to help you to kind of understand this better. So what I have here, okay, with us, is basically a square prism. Okay, so you can see that there's a square there at the top, there's a square at the bottom. Okay, so if we take the area of this square plus the area of this square, that takes care of the two bases. But now if you can kind of see here, see the perimeter of this square. Okay, so you're following me. So if you take that perimeter, if you unfold this figure, that's the perimeter of that base. Okay, remember this was folded up like so, okay, but when we unfold it, you're going to see that the perimeter of this square is the same as the perimeter of this That's the perimeter of our base times the height. And when you take the length times the width or the base times the height, you're getting the area of this rectangle. Okay, so you're with me so far. So if I was to draw a net, okay, net is like if you were to unfold the shape, what you would get is something that looks like this. You would get four rectangles and you would have a square here and a square here. It's like taking a shoebox and unfolding it. You might want to try that. And so basically, this dimension right here is the perimeter of the base. This dimension here is the height, and that's where we get the p times h. That's what gives us the lateral area, or the area of the sides. Then you just add the two bases, the top and bottom. Okay, so let's do this. So basically the perimeter is going to be 4 plus 4 plus 4 plus 4, which is 16, times the height, which is 10. Okay, so on my little diagram here, this would be 10, and this would be 16, right? So that's 160. Okay, plus the two bases. Now the base is a square, so that's 4 times 4 is 16, times 2, since we have two of them. So we have 32. plus 160 which is 192 inches squared. Okay now when you do area it's square units because you're covering the surface with these little one by one squares. Okay and that's how many would cover the entire outer surface. Okay so you're with me so far? So now let's go over to this shape here. Try this one on your own if you can and basically what we're going to do is do the same thing. This is a triangular prism. The top and bottom are triangles. They're separated by that height which you can see is 12. So let's start with the volume first. The volume is the area of the base times the height. Okay, now because this is a triangle, what you want to do is you want to think of the area formula for a triangle. That's one half little b times h. So this is like our base here, three. This is like our height of our triangle here, four. So this is going to be one half times three times four, and then our height is going to be 12. Okay, so You don't want to get the heights confused. This h is for the overall height, the distance between the two triangles. This h is really for the height of our triangle. So that's why I recommend just memorizing the general formula. Then you can break it down into the sub-formulas for each individual. problem. So let's go ahead and simplify this. So this comes out to, let's see, 3 times 4 is 12, times 12 is 144, half of that is 72, and I didn't put the units here, so I'm just going to say units cubed for the volume, all right? Now let's go over to the surface area. So surface area is really the total outer area, right? So what we're going to do is we're going to use the same formula we were talking about here earlier, okay, which is the 2b plus ph, so two bases plus the perimeter of the base times the height. So in this case, the base is going to be a triangle. So that's going to be 2 times 1 half base times height, plus the perimeter of the base times the overall height. So let's see if we can do this. So basically, 2 times 1 half, that's 1. So I'm just going to cancel those out. The base of the triangle is 3, and the height of the triangle is 4. The perimeter of the base, you have to add all these up. 3 plus 4 plus 5, which gives us 12. And the overall height is 12. So we get 144. plus 12, which is 156 units squared, because this is area. So we're just covering this with these little squares. Now, if you want to see a net, okay, like an unfolded picture of this diagram, let me see if I can put that here for us. It would look something like this. Okay. You would have three, four, and five. Okay. So when you unfold this, and then you would have a triangle over here like this and a triangle over here like this. So if when you fold those, that's going to be the top and the bottom. So that's where we're getting this 12. That's the perimeter. Okay times the height which is this height here That's 12. So that's how I'm getting 12 times 12, which gives you this area 144 then you just have to add the two bases the two triangles to get the total area Okay one more now we're going to talk about cylinders again a cylinder is like a circular prism You have the top and bottom are circles. They're separated by that height. So let's start with the volume So the volume is the area of the base times the height, right? So notice I'm using the same formula for all these Prisms cylinders we're treating them as a group. Okay, so the area of the base. It's a circle So I'm gonna write the sub formula for the base, which is pi R squared times the height So in this case the radius of the circle is 3 so that's going to be 3 squared The overall height is 6 so if we do this, this is 3 squared is 9 times 6 is 54 Pi you can put pi in your calculator 3.14 and multiply these together, but I'm just gonna leave it as 54 pi meters cubed and again, that's how many little ice cubes that are one meter by one meter by one meter that would fill this gigantic cylinder. Now, if we want to do the surface area, okay, what we're going to do is we're going to, again, think of this as a net. So if we unfold this, like if I was to take some scissors and cut that right there and unfold it, you're going to have a circle here, a circle here, and you're going to have a rectangle here. When you roll that rectangle, let me see if I can show you with a piece of paper here. It's going to look like this, right? So there's the circle at the top, there's the circle at the bottom. But when I unroll this, there's our rectangle right there. Now when you look at this dimension here, this dimension is really the circumference of the circle, right? So when I unroll that, this is the circumference. That's just another name for the perimeter of the base. So we're going to use our formula 2B plus pH. But remember, the perimeter is really this dimension. Okay, that's like unrolling this circle here like this. So this is going to give us 2 pi r. This is our height. So that's where we get our perimeter, 2 pi r times the overall height plus two bases. Remember, the base is a circle, so pi r squared. So now all we have to do is substitute in the values. So we've got 2 pi, the radius is 3 squared, plus 2 pi, the radius is 3, and the height is 6. And we just have to simplify. So this would be 9 times 2 is 18 pi. This is 18 times 2 is 36 pi. If we add those together, we get 54 pi meters squared. And it's just a coincidence that these both came out to 54 pi. This is meters squared. This is how many square meters would cover the outer surface of this cylinder. So again, think about volume and surface area, you know, just these three basic formulas, b times h, the area of the base times the height. And then the surface area formula 2b plus pH to find the outer area. Okay, next we're going to talk about pyramids and cones. Let me erase this board and we'll start with those. Okay, now we're going to talk about pyramids and cones. And you want to group pyramids and cones together just to help you to narrow down the number of formulas you have to memorize and just make it easier overall. So a pyramid and a cone, you'll notice they both just have one base. So they just have the one bottom, unlike prisms that had a top and a bottom that were parallel and congruent. These just have one base and you can see they go up to a point here. Okay, this one vertex point at the top. Now when you look at a pyramid and a cone they've got two different things going on. They've got this overall height, okay, which goes right down The center of the base see this 12 right here And then they have something called the slant height or which they use the letter L for Which I kind of call it the leaning height because it's not an angle like that So that's the slant height right there 13 and in this case the slant height is 5 So we're going to talk about those in these problems, but first let's do the volume So the volume we're going to use this formula here 1 3rd the area of the base times the height, okay? So what you do is you take the area of this bottom piece which is 6 times 6 that's 36 times 1 3rd times the height. Now the height that you want to use is this overall height straight down to the center of the base. In that case this one's four so this comes out to let's see 48 inches cubed since it's volume. All right so you're with me so far? What this means this one-third area of the base times the height remember when we did the prisms okay like if I was to draw a prism like say for example like this okay just a rough sketch here what you can see is if these had the same base and they had the same height. If you put this pyramid inside of this prism, you would actually be able to fit three of these inside of there. Okay, so this one plus two more, because you can see it's tapered, it's, you know, angled up to a point like so. So this actually only takes up a third of this volume. So that's where this one third is coming into play. Now, let's go over to this one. Let's talk about the cone now. So same formula, volume is one third, area of the base times the height. In this case, the base is a circle, so we're going to use the formula pi r squared for a circle, okay, area of a circle, times the height. And again, we want to use this overall height straight down to the center of the base, that's 12. And let's see, so for the circle, the radius is 5, so that's going to be 5 squared is 25. Okay, now just a note here to you, and that's that, you know, when you're... Multiplying these together, multiplication is commutative. You can change the order, you're going to get the same result. So I'm going to take 1 third of 12, which is 4, times 25 is 100, times pi. So this is going to be 100 pi units cubed. And that's it, you got the volume. Now let's talk about the surface area. So the surface area, the formula we're going to use for both the pyramid and the cone is area of the base, since you just have one bottom, okay, plus one half the perimeter of the base times the slant height. Okay, and it's the same formula over here. Volume is the area of the base plus one half the perimeter times the slant height. But what's different is it's a different shape base. See, this is a square, so 6 times 6 is 36, right? This one over here is a circle, so that's going to be pi r squared, so pi times 5 squared. Okay, over here, we're going to take the perimeter, which is 6 plus 6 plus 6 plus 6, 4 6's, which is 24, and the slant height or the leaning height, which is 5, right? But over here, what we have is we have a circle, so we have to take the perimeter of our circle, which is the circumference, which, remember, the formula for circumference is 2 pi r, so 2 pi times 5. times the slant height, so that leaning height, which is 13. So when I think of the L, I think of the leaning height, right? Now all we have to do is go back and simplify. So we've got half of 24, which is 12. 12 times 5 is 60. And we just add the 36 and the 60 together to get 96 inches squared. So remember, square for area, that's two-dimensional. Cubed for volume, that's three-dimensional. For this one, we get 5 squared is 25 pi. Let's see, 2 times 5 is 10, times 1 half is 5, that's 5 pi, times 13 is 65 pi, right? And if we add those together, what do we get? We get 80, 90 pi units squared. Now notice I just left the pi in there, that's an exact answer, but if you want to get an approximation, you can put 3.14 for pi, multiply it by 90, and you got it. So again, for pyramids and cones, you want to think of the two basic formulas. which is the volume formula. See, it's the same, one-third area of the base times the height. And for the surface area, I wrote volume here, this is surface area, okay, the outer surface, okay, one base plus one-half PL. So it's just the area of the one bottom plus one-half the perimeter times the slant height. So let me erase this board, and then we're going to talk about spheres. But the key is to group these together to make it easier, you know, to memorize these formulas. Okay, lastly, we're going to talk about spheres. But before I get into spheres, I just wanted to mention that if you're preparing for the ACT or the SAT math section and you want to boost your score, I've got two courses available. You can check them out. I've got links on my About page. I'll put links in the description below. But basically, I've got one for the SAT, which goes over 39 concept areas that are important to really boost your score on the test. And then with the ACT math section, I've got 65 concepts. They're really going to depth with teaching and examples and practice problems and so forth to really help you to boost your score. So if you like my teaching style, check out those courses. A lot of students have benefited from them and I'm sure you will too. But let's talk about spheres. now so a sphere is basically like a circle in three dimensions you know it's the set of all points that are equidistant from a given point which is called the center so it's basically like a ball like a basketball right and so if we're trying to find the volume like the inner space the three-dimensional space we're going to be using this formula here it's four-thirds pi times the radius cubed so there's really just that one dimension the radius now sometimes I'll try to confuse you a little bit maybe by giving you the diameter but you can always cut it in half to get the radius So for this one, we're just going to say 4 3rds pi times 3 cubed. 3 cubed is 3 times 3 times 3, which is 27. So we have 4 3rds times pi times 27, which is like 27 over 1. And you can do a little bit of cross-reducing numerator and denominator here. So that ends up coming out to multiply the numerators and denominators together. That gives you 36 pi units cubed. That's the volume, right? Now for the surface area, the formula is a little bit different. It's 4 pi r squared. Okay, and what we want to do here is just put in our radius, which is 3. So that's 3 squared is 9 times 4 is 36 pi units squared since it's area. That's the outer surface. Now, it's just a coincidence we came out with 36 for both of these, but the formula is different. You won't always get that situation. So, again, just trying to show you an easy way to group these together. memorize the general formula and then you can break it down for each specific shape whether it's a triangular pyramid or it's a square pyramid or it's a cone or whatever type of shape it is. So I hope you enjoyed this video. Subscribe to the channel. Check out over the over 400 other videos I have on my Mario's Math Tutor YouTube channel to help you boost your score in your math class, improve your understanding, and make learning math a lot less stressful. And I look forward to seeing you in the future videos. I'll talk to you soon.

So let's get into the first group of problems. The first group, we're talking about prisms. And basically what a prism is, Prism is a figure, a three-dimensional figure, where the two bases are parallel, okay, and they're congruent.

So that means they're the same size and shape, and they're parallel, they don't cross, okay, if they were to continue to go, and they're separated by the height, okay. Now I group cylinders, okay, with these prisms because a cylinder is really like a circular prism. You can see the two bases are circles, and they're separated by that height. So when you think about cylinders and prisms, think of them all as one group.

So we're going to do the surface area and the volume. So first of all, let's do the volume of this figure. Now when you do the volume, the formula, the general formula, is very easy to remember. It's the area of the base times the height.

So that's why I did a capital B. So the area of the base means you find the area of the bottom, that's what it sits on, and you multiply by the height. So essentially what this is like, it's like taking the area of this square in this case, okay, which is 16 inches squared. And it's like taking those squares, like a stack of sticky notes, and you stack them up, okay, 10 inches high. So you take the area of the base, which is 16, times the height, which is 10, which is going to give us 160 inches cubed.

Now, the reason it's cubed is because, you know, we're filling these up with, like, little ice cubes you could think of, okay, something like that, right? So little one-by-one-by-one cubes, and that's how many are going to fill up the inside of this box, right? Now let's talk about the surface area for a moment.

So the surface area is like if you were going to take a paintbrush and you were going to paint all the sides, you know, of this figure. So when we think of the surface area, this is the formula that you want to memorize. It's 2B, okay, the capital B means the area of the base, two of them because you have a top and a bottom, plus the perimeter of the base times the height.

Now let me see if I can show you a three-dimensional figure to help you to kind of understand this better. So what I have here, okay, with us, is basically a square prism. Okay, so you can see that there's a square there at the top, there's a square at the bottom. Okay, so if we take the area of this square plus the area of this square, that takes care of the two bases. But now if you can kind of see here, see the perimeter of this square.

Okay, so you're following me. So if you take that perimeter, if you unfold this figure, that's the perimeter of that base. Okay, remember this was folded up like so, okay, but when we unfold it, you're going to see that the perimeter of this square is the same as the perimeter of this That's the perimeter of our base times the height.

And when you take the length times the width or the base times the height, you're getting the area of this rectangle. Okay, so you're with me so far. So if I was to draw a net, okay, net is like if you were to unfold the shape, what you would get is something that looks like this.

You would get four rectangles and you would have a square here and a square here. It's like taking a shoebox and unfolding it. You might want to try that. And so basically, this dimension right here is the perimeter of the base.

This dimension here is the height, and that's where we get the p times h. That's what gives us the lateral area, or the area of the sides. Then you just add the two bases, the top and bottom. Okay, so let's do this.

So basically the perimeter is going to be 4 plus 4 plus 4 plus 4, which is 16, times the height, which is 10. Okay, so on my little diagram here, this would be 10, and this would be 16, right? So that's 160. Okay, plus the two bases. Now the base is a square, so that's 4 times 4 is 16, times 2, since we have two of them.

So we have 32. plus 160 which is 192 inches squared. Okay now when you do area it's square units because you're covering the surface with these little one by one squares. Okay and that's how many would cover the entire outer surface. Okay so you're with me so far? So now let's go over to this shape here.

Try this one on your own if you can and basically what we're going to do is do the same thing. This is a triangular prism. The top and bottom are triangles. They're separated by that height which you can see is 12. So let's start with the volume first.

The volume is the area of the base times the height. Okay, now because this is a triangle, what you want to do is you want to think of the area formula for a triangle. That's one half little b times h. So this is like our base here, three. This is like our height of our triangle here, four.

So this is going to be one half times three times four, and then our height is going to be 12. Okay, so You don't want to get the heights confused. This h is for the overall height, the distance between the two triangles. This h is really for the height of our triangle. So that's why I recommend just memorizing the general formula. Then you can break it down into the sub-formulas for each individual.

problem. So let's go ahead and simplify this. So this comes out to, let's see, 3 times 4 is 12, times 12 is 144, half of that is 72, and I didn't put the units here, so I'm just going to say units cubed for the volume, all right? Now let's go over to the surface area. So surface area is really the total outer area, right?

So what we're going to do is we're going to use the same formula we were talking about here earlier, okay, which is the 2b plus ph, so two bases plus the perimeter of the base times the height. So in this case, the base is going to be a triangle. So that's going to be 2 times 1 half base times height, plus the perimeter of the base times the overall height.

So let's see if we can do this. So basically, 2 times 1 half, that's 1. So I'm just going to cancel those out. The base of the triangle is 3, and the height of the triangle is 4. The perimeter of the base, you have to add all these up.

3 plus 4 plus 5, which gives us 12. And the overall height is 12. So we get 144. plus 12, which is 156 units squared, because this is area. So we're just covering this with these little squares. Now, if you want to see a net, okay, like an unfolded picture of this diagram, let me see if I can put that here for us. It would look something like this. Okay.

You would have three, four, and five. Okay. So when you unfold this, and then you would have a triangle over here like this and a triangle over here like this.

So if when you fold those, that's going to be the top and the bottom. So that's where we're getting this 12. That's the perimeter. Okay times the height which is this height here That's 12. So that's how I'm getting 12 times 12, which gives you this area 144 then you just have to add the two bases the two triangles to get the total area Okay one more now we're going to talk about cylinders again a cylinder is like a circular prism You have the top and bottom are circles. They're separated by that height. So let's start with the volume So the volume is the area of the base times the height, right?

So notice I'm using the same formula for all these Prisms cylinders we're treating them as a group. Okay, so the area of the base. It's a circle So I'm gonna write the sub formula for the base, which is pi R squared times the height So in this case the radius of the circle is 3 so that's going to be 3 squared The overall height is 6 so if we do this, this is 3 squared is 9 times 6 is 54 Pi you can put pi in your calculator 3.14 and multiply these together, but I'm just gonna leave it as 54 pi meters cubed and again, that's how many little ice cubes that are one meter by one meter by one meter that would fill this gigantic cylinder.

Now, if we want to do the surface area, okay, what we're going to do is we're going to, again, think of this as a net. So if we unfold this, like if I was to take some scissors and cut that right there and unfold it, you're going to have a circle here, a circle here, and you're going to have a rectangle here. When you roll that rectangle, let me see if I can show you with a piece of paper here. It's going to look like this, right?

So there's the circle at the top, there's the circle at the bottom. But when I unroll this, there's our rectangle right there. Now when you look at this dimension here, this dimension is really the circumference of the circle, right?

So when I unroll that, this is the circumference. That's just another name for the perimeter of the base. So we're going to use our formula 2B plus pH. But remember, the perimeter is really this dimension.

Okay, that's like unrolling this circle here like this. So this is going to give us 2 pi r. This is our height. So that's where we get our perimeter, 2 pi r times the overall height plus two bases.

Remember, the base is a circle, so pi r squared. So now all we have to do is substitute in the values. So we've got 2 pi, the radius is 3 squared, plus 2 pi, the radius is 3, and the height is 6. And we just have to simplify. So this would be 9 times 2 is 18 pi.

This is 18 times 2 is 36 pi. If we add those together, we get 54 pi meters squared. And it's just a coincidence that these both came out to 54 pi.

This is meters squared. This is how many square meters would cover the outer surface of this cylinder. So again, think about volume and surface area, you know, just these three basic formulas, b times h, the area of the base times the height.

And then the surface area formula 2b plus pH to find the outer area. Okay, next we're going to talk about pyramids and cones. Let me erase this board and we'll start with those.

Okay, now we're going to talk about pyramids and cones. And you want to group pyramids and cones together just to help you to narrow down the number of formulas you have to memorize and just make it easier overall. So a pyramid and a cone, you'll notice they both just have one base.

So they just have the one bottom, unlike prisms that had a top and a bottom that were parallel and congruent. These just have one base and you can see they go up to a point here. Okay, this one vertex point at the top. Now when you look at a pyramid and a cone they've got two different things going on. They've got this overall height, okay, which goes right down The center of the base see this 12 right here And then they have something called the slant height or which they use the letter L for Which I kind of call it the leaning height because it's not an angle like that So that's the slant height right there 13 and in this case the slant height is 5 So we're going to talk about those in these problems, but first let's do the volume So the volume we're going to use this formula here 1 3rd the area of the base times the height, okay?

So what you do is you take the area of this bottom piece which is 6 times 6 that's 36 times 1 3rd times the height. Now the height that you want to use is this overall height straight down to the center of the base. In that case this one's four so this comes out to let's see 48 inches cubed since it's volume. All right so you're with me so far? What this means this one-third area of the base times the height remember when we did the prisms okay like if I was to draw a prism like say for example like this okay just a rough sketch here what you can see is if these had the same base and they had the same height.

If you put this pyramid inside of this prism, you would actually be able to fit three of these inside of there. Okay, so this one plus two more, because you can see it's tapered, it's, you know, angled up to a point like so. So this actually only takes up a third of this volume. So that's where this one third is coming into play. Now, let's go over to this one.

Let's talk about the cone now. So same formula, volume is one third, area of the base times the height. In this case, the base is a circle, so we're going to use the formula pi r squared for a circle, okay, area of a circle, times the height. And again, we want to use this overall height straight down to the center of the base, that's 12. And let's see, so for the circle, the radius is 5, so that's going to be 5 squared is 25. Okay, now just a note here to you, and that's that, you know, when you're...

Multiplying these together, multiplication is commutative. You can change the order, you're going to get the same result. So I'm going to take 1 third of 12, which is 4, times 25 is 100, times pi. So this is going to be 100 pi units cubed.

And that's it, you got the volume. Now let's talk about the surface area. So the surface area, the formula we're going to use for both the pyramid and the cone is area of the base, since you just have one bottom, okay, plus one half the perimeter of the base times the slant height.

Okay, and it's the same formula over here. Volume is the area of the base plus one half the perimeter times the slant height. But what's different is it's a different shape base.

See, this is a square, so 6 times 6 is 36, right? This one over here is a circle, so that's going to be pi r squared, so pi times 5 squared. Okay, over here, we're going to take the perimeter, which is 6 plus 6 plus 6 plus 6, 4 6's, which is 24, and the slant height or the leaning height, which is 5, right?

But over here, what we have is we have a circle, so we have to take the perimeter of our circle, which is the circumference, which, remember, the formula for circumference is 2 pi r, so 2 pi times 5. times the slant height, so that leaning height, which is 13. So when I think of the L, I think of the leaning height, right? Now all we have to do is go back and simplify. So we've got half of 24, which is 12. 12 times 5 is 60. And we just add the 36 and the 60 together to get 96 inches squared.

So remember, square for area, that's two-dimensional. Cubed for volume, that's three-dimensional. For this one, we get 5 squared is 25 pi.

Let's see, 2 times 5 is 10, times 1 half is 5, that's 5 pi, times 13 is 65 pi, right? And if we add those together, what do we get? We get 80, 90 pi units squared.

Now notice I just left the pi in there, that's an exact answer, but if you want to get an approximation, you can put 3.14 for pi, multiply it by 90, and you got it. So again, for pyramids and cones, you want to think of the two basic formulas. which is the volume formula.

See, it's the same, one-third area of the base times the height. And for the surface area, I wrote volume here, this is surface area, okay, the outer surface, okay, one base plus one-half PL. So it's just the area of the one bottom plus one-half the perimeter times the slant height. So let me erase this board, and then we're going to talk about spheres.

But the key is to group these together to make it easier, you know, to memorize these formulas. Okay, lastly, we're going to talk about spheres. But before I get into spheres, I just wanted to mention that if you're preparing for the ACT or the SAT math section and you want to boost your score, I've got two courses available.

You can check them out. I've got links on my About page. I'll put links in the description below.

But basically, I've got one for the SAT, which goes over 39 concept areas that are important to really boost your score on the test. And then with the ACT math section, I've got 65 concepts. They're really going to depth with teaching and examples and practice problems and so forth to really help you to boost your score. So if you like my teaching style, check out those courses.

A lot of students have benefited from them and I'm sure you will too. But let's talk about spheres. now so a sphere is basically like a circle in three dimensions you know it's the set of all points that are equidistant from a given point which is called the center so it's basically like a ball like a basketball right and so if we're trying to find the volume like the inner space the three-dimensional space we're going to be using this formula here it's four-thirds pi times the radius cubed so there's really just that one dimension the radius now sometimes I'll try to confuse you a little bit maybe by giving you the diameter but you can always cut it in half to get the radius So for this one, we're just going to say 4 3rds pi times 3 cubed. 3 cubed is 3 times 3 times 3, which is 27. So we have 4 3rds times pi times 27, which is like 27 over 1. And you can do a little bit of cross-reducing numerator and denominator here.

So that ends up coming out to multiply the numerators and denominators together. That gives you 36 pi units cubed. That's the volume, right?

Now for the surface area, the formula is a little bit different. It's 4 pi r squared. Okay, and what we want to do here is just put in our radius, which is 3. So that's 3 squared is 9 times 4 is 36 pi units squared since it's area.

That's the outer surface. Now, it's just a coincidence we came out with 36 for both of these, but the formula is different. You won't always get that situation.

So, again, just trying to show you an easy way to group these together. memorize the general formula and then you can break it down for each specific shape whether it's a triangular pyramid or it's a square pyramid or it's a cone or whatever type of shape it is. So I hope you enjoyed this video. Subscribe to the channel. Check out over the over 400 other videos I have on my Mario's Math Tutor YouTube channel to help you boost your score in your math class, improve your understanding, and make learning math a lot less stressful.

And I look forward to seeing you in the future videos. I'll talk to you soon.

Transcript for:Understanding Surface Area and Volume

Transcript for:
Understanding Surface Area and Volume