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Volume and Surface Area
Volume and surface area. In all the previous geometry that we’ve talked about, all the lessons up to now, we’ve been looking at 2D figures. That is to say flat things, that would just be drawn on a two-dimensional surface. Now we’re going to look a little at three-dimensional geometry.
In 2D we can talk about the area of shapes. In three dimensions, we’ll also talk about the volume, of shapes. We’ll start with the easiest 3D shape the cube. The cube of course is very special, much in the same way that a square is special among quadrilaterals. The cube is special among 3D shapes in may ways.
So what’s true about a cube? Well first of all, it has six faces, and these six faces are six congruent squares, it has eight vertices, that is to say, eight corners, it has 12 edges, 12 congruent edges. Those edges of course are the, the little line segments that connects vertex to vertex.
And it is true, that the faces meet at right angles at the edges and three mutually perpendicular edges, meet at each vertex. So these are the properties of a cube, very special. If a cube has an edge-length S, then its volume, is given by, V equals S cube. When we raise a number to the third power, we call this cubing the number, precisely because this is how we find the volume of the cube.
So, both the algebra term to square a number and the algebra term to cube a number, these are terms that have their historical roots in geometry. Each face on this cube is a square with side S and so it would have an area of s squared and that means that the total, because there are six faces. The total surface area, of a cube would be 6s squared. The cube is a special case, of a more general category known as rectangular solids.
The sides can be an rectangles, the faces. But we still have the requirement that the faces meet at right angles at each edge and three mutually perpendicular edges meet at each vertex. So most boxes and cartons, are rectangular solids. They’re, the world is just absolutely full of rectangular solids. The volume of a rectangular solid, is the product of the three different edge lengths.
So if we call those, those lengths height, width and depth. The volume is just the product of those three. The surface area, well this is tricky. We can pair the rectangles, because of course opposite faces are congruent. So we have two rectangles that are h times w, that’s the front and the back. We have h times d, that’s the right and the left side.
And then, w times d, that’s the top and the bottom. And, so, then, we add all those up for the total surface area. I would urge you not to memorize this as a formula. I would urge you just to think through the process, of finding the areas of those rectangles and adding them up. In a rectangular solid, we can consider two different kinds of diagonals.
One kind, called a face diagonal, is a diagonal of only one face of the solid. And of course, we can always find this length, with the Pythagorean Theorem. We just use the, the two edge lengths. Those are of course the two legs of the triangle. So here we have a three four five triangle and AC equals five. The other kind of diagonal called a space diagonal does not run along a face, it passes through the interior of a rectangular solid from one vertex to the opposite vertex.
So this is a little harder to visualize. This is going from vertex A, through the center of the figure, and coming out at the opposite vertex at D. And so, for this, we use a three-dimensional version of the Pythagorean theorem. So it’s actually, edge squared plus edge squared plus edge squared, as only those three edges are all perpendicular to each other.
Then we can square all three of them, add them together and that will equal the square, of the space diagonal. So here we get, 3 squared 9, 4 squared 16, 5 squared 25, add them up we get 50. So that means that AD is the square root of 50. And of course we can simplify that, to 5 root 2. And so 5 root 2 is the length of that space diagonal.
Now the test is not probably not gonna use the term, space diagonal. But it will give you that, for example when I give you that diagram and say find the length of AD. And so you’re gonna have to know, how to do this. What is the length of the space diagonal of a cube, of edge length S? Well again we’ll use the 3D Pythagorean Theorem here.
And, of course, we have three edges, each with a length of s. So, AB squared is just gonna be s squared + s squared + s squared, which, of course, is 3s squared. Take a square root, and what we get is that AB, is the cube root, is the square root of 3 times s. So, this is interesting.
The diagonal of a square, is root two times the side, the space diagonal of a cube is root three times the side. Very interesting. And the test loves that little fact. Here’s a practice problem.
Pause the video, and then we’ll talk about this. Okay, so we’re given two of the three edge lengths. We have the four and the six. We don’t know, the length from front to back. We’re missing an edge length there.
But we are given the length of the space diagonal JK. And we need to find the volume. So the first that we need, is we need to find that missing edge length. So call that d, the depth, we don’t know that. Four squared, plus 6 squared, plus d squared equals eight squared. And so four squared plus six squared that adds up to 52, plus d squared equals 64.
D squared equals 12, take the square root, d equals the square root of 12. We can simplify that to 2 root 3. If all this stuff about simplifying square roots is unfamiliar to you, I would strongly suggest go back and watch the video on powers and roots, on simplifying square roots. That would be very important to appreciate.
Before you understand all the square roots in this lesson. Well, now, we have our missing edge length. So now, we know all three edges. So, of course, the volume is just the product of those edges. Multiply this together, and we get 48, root 3, and that is the volume. Another 3D shape that the test could expect you to know, is the cylinder.
The cylinder, when resting on the circular base, has a height of h. The radius of each circular base is r. So they’re two congruent circles at the top and the bottom and then this curved side connecting them. That’s a cylinder. So the volume, you can think of it as just the base times the height and of course, the base is a circle.
So that’s pi r squared, the area of the base times the height. Pi r squared, h. That’s the volume of a cylinder. The total surface area of a cylinder is interesting. Obviously, the circular top and bottom are easy. Each one has an area of pi r squared.
That’s the easy part. Think about the curved side, what is called the, lateral, surface area of a cylinder. Think about the paper label on a metal can. If we cut the label vertically, we can unroll it into a flat rectangle. So imagine that we do that.
We cut that lateral side and unroll it into a rectangle. And so the the whole area of the surface area of the cylinder would look something like this. We have the circle at the top and the bottom. Those are the, the Pi r squares, we know that. And then we have that long rectangle.
Now think about that rectangle. The top edge of that long rectangle. So, the edge right here. That had to run around, that fit neatly around the top circle. And so, what that means is that, that edge must be exactly as long as the distance around that circle.
And, of course, the distance around the circle, that’s the circumference, pi, 2 pi r, so it means that the top edge of that rectangle is 2 pi r. Well, now we know, we have the height already as h, we know the base and the height of the rectangle, so that lateral area, the area of that rectangle, base times height, 2 pi r h. And then we can add that to the two circles.
The total surface area is pi r squared plus 2 pi rh. Once again, I will ask you not to memorize that formula. Please understand the logic that we talked about, and why this formula is true. The only other 3D shape I will mention is the sphere. You don’t need to memorize anything it. You simply need to know what it is.
So here’s a picture of a sphere. Every point on the surface is equidistance from the center. The sphere is circular in every direction. And one way to think about it is the surface of the Earth is roughly a sphere. So you could walk around, obviously you can’t walk all the way around the Earth, but, for example, you could fly, you could fly all the way around the Equator, you could fly over the, the Prime Meridian, the North Pole, then down around the South Pole.
So there are a bunch of circles you could make in every direction, and you’d always come back to the same place. Here’s a practice problem, pause the video and then we’ll talk about this. A cylinder neatly encloses a sphere so that the curve of the cylinder touches the sphere in a circle at its widest, and the top & bottom faces of the cylinder are tangent to the sphere.
If the volume of the sphere is four thirds pi r cubed. What fraction of the cylinder does the sphere occupy? Hm. Well this one requires a lot of visualization. Let’s think about this. First of all, the radius of the sphere and the radius of the cylinder must be the same.
That’s how they touch each other all the way around that circle at the middle. So they have the same radius, we’ll just call that r. The height of the cylinder must be as tall as a sphere. So in other words, it’s height must be the diameter of the sphere. And of course that diameter would be 2r. So h equals 2r.
So now the, the volume of the cylinder, course it’s pi r squared h, or pi r squared 2 r and we just simplify that to 2 pi r cubed. Well now we’re gonna take a ratio. The sphere is taking up what, what fraction of the cylinder. Well we’re gonna divided four, four-third pie r squared of course the pie r squareds will cancel and we’ll just be left with the number.
Four-thirds divided by two or four-thirds times one half and that is two-thirds and that’s the fraction that the sphere would take up of the cylinder. That fact was very important to Mr. Archimedes, he actually put it on his tombstone. In summary, for the cube we have a formula for the volume, V equals S cubed.
And for the total surface area, 6S squared. For rectangular solid, it’s important to know how to find the volume as well as the surface area. And for a cylinder, it’s very important to find the volume as well as the surface area.
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