>> From time to time, we'll be talking about a variety of three-dimensional figures. The first one we'll talk about here is the sphere. And we'll talk about the equation associated with a sphere. And it comes from the equation associated with a circle. Now, remember that when the circles were developed and the equations for circles were developed, they were developed in terms of the distance formula. If we have a circle whose center is at 4, 3, and it has a particular radius, then you use the distance formula by thinking of the distance, R. And the distance is between the center of the circle and a general point X, Y that is on the circle itself. The distance formula is this. And that distance is the distance, R, that we're talking about here. A difference in Xs is the general X on the circle minus the X component for the center. And then the difference in Y is the Y coordinate on the circle and the Y coordinate at the center. And then using this, we square on both sides to clear the radical. And we have R squared is equal to this plus this. And to generalize the thing further, the situation further, if we're thinking about a center at a general point, H, K, then we have an H here and a K here. So the center is at H, K, and the radius is R. This is the equation of a circle. Okay, now, we just extend the idea for the equation for a sphere. And we extend it just utilizing another point in the process into this other dimension, into this Z dimension. So it's the distance is equal to, and we start with that distance formula, and then that evolves into a situation where we have an R squared here, the squaring clears the radical once again, and it's X minus H, Y minus K, Z minus L. And the center of the sphere is at H, K, L. You see, that's the idea. And the radius is R. Well, let's work a few problems. Suppose we have a sphere and the center is located at 2, 4, 3. At radius is 3. And we want to write the equation in standard form. And the equation can take a lot of forms. But the standard form is the form that we just looked at. So we might begin with the skeleton of the equation in standard form and just fill in the information we know. And we know just about everything here. We have the H, K, L is up here. So H is replaced with 2 and K is replaced with 4 and L is replaced with 3. The radius is 3. So R is replaced with 3. And we would presumably, though, probably write this as the 3 squared is 9 and this is the equation that we're looking for.