Our last video in the geometry series deals with three new 3D shapes. We will learn here about cylinders, cones and spheres.
Transcript: Cylinders, Cones and Spheres
The test could also ask about 3D shapes that have curved faces. And these would be cylinders, cones, and spheres. I’ll just point out that everything in this video was discovered by one individual. It was the great Archimedes, the ancient Greek mathematician–he figured out all of this.
And here’s Mr. Archimedes’ picture on the Field Medal, which is sort of the Nobel Prize of mathematics.
So first of all, let’s talk about cylinders. The cylinder, when resting on one circular base, has a height of h. The radius of each circular base is r. So it’s two congruent circles and they’re connected by this curve thing.
You could say that cylinders, in some ways, are circular versions of a prism. It’s important to know the volume of cylinders. Pi r squared h, the test could expect you to know that. The total surface area of a cylinder is interesting. Obviously, the circular top and bottom are easy, each has an area of pi r squared. Think about the curved side, what is called the lateral surface of the cylinder.
Think about a paper label on a metal can, if we cut the label vertically, we can unroll it into a flat rectangle. So imagine, we have the top, bottom of a can, and then that label. We cut it vertically and then we unrolled it, so it was flat. The top edge of that long rectangle fits around the top of the circle. So that means that the length of that is the circumference of the circle, 2 pi r.
And of course, the height is h. So we have a height, h, and a width, 2 pi r. So the lateral surface is 2 pi rh, that’s the rectangle. Add the two circles, we get 2 pi r squared + 2 pi rh. The test does not expect you to know this. You do not have to memorize that, but the test may ask about it. They’ll give you that formula, it is good to understand where that formula comes from.
And so it’s just good to be familiar with this in case the test hands it to you.
Now we’ll talk about cones. A cone is like a circular pyramid. It has a circular base and a surface formed by straight line segments from each point on the circumference up to a single vertex. And so that makes this slanted, curving surface that goes up to the point.
If cones appear at all, the test will give you the formula for the volume of a cone, and that formula is V = one-third pi r squared h. The test will not ask at all about the curved lateral surface of a cone. That’s very difficult, so don’t worry about that.
Finally, spheres. Here you need to know just the basic facts.
You need to know that every point on the surface is equidistant from the center. You need to know that the sphere is circular in every direction. Incidentally, the surface of the Earth is roughly a sphere, not exactly. It turns out that the Earth is a little wider at the equator, but it’s roughly a sphere. The test will give you any formula you need to know.
For example, the volume of a sphere or the surface area of a sphere. So you don’t need to memorize those, the test will give those to you.
Here’s a practice problem, pause the video and then we’ll talk about this.
Okay, cylinders neatly enclose a sphere, so that the curve of the cylinder touches the sphere in a circle at its widest. Also, the top and bottom faces of the cylinder are tangent to the sphere.
If the volume of a cylinder is four-thirds pi r squared, what fraction of the cylinder does the sphere occupy? Okay, well, let’s think about this. The radius of the sphere and that of the cylinder must be the same, r, because they touch each other. So that’s the radius.
The height of the cylinder must be as tall as the sphere, so it must equal the diameter, which is 2r. The height of this cylinder is 2r. That means that this cylinder, we could figure out its volume, 2 pi rh. So that’s 2 pi r squared times 2r. 2 r cubed, that’s the volume of the cylinder.
So if we take the volume of the sphere, four-thirds pi r cubed, and divide by 2 pi r cubed, the pi r cubed parts will cancel and we’ll just be left with four-thirds divided by 2. That’s four-thirds times one-half, or in other words, two-thirds. So the sphere takes up two-thirds of the cylinder.
In summary, for cylinders, you need to know the volume of the cylinder.
If they ask about the surface area, they will give you that formula. For cones, if they ask about the volume, they will give you that formula. Spheres, you need to know what a sphere is. If the test makers ask about either the volume or the surface area, they will provide the necessary formula.