Features of a circle from its graph | Mathematics II | High School Math | Khan Academy

So we have a circle right over here. The first question we'll ask ourselves is: what are the coordinates of the center of that circle? Well, we can eyeball that. We can see it looks like the center is centered on that point right over there. The coordinates of that point, the x-coordinate is -4 and the y-coordinate is -7. So the center of that circle would be the point (-4, -7).

Now, let's say on top of that, someone were to tell us that this point (-5, 9) is also on the circle. So, (-5, 9) is on the circle. Based on this information—the coordinate of the center and a point that sits on the circle—can we figure out the radius? Well, the radius is just the distance between the center of the circle and any point on the circle. In fact, one of the most typical definitions of a circle is all of the points that are the same distance, or that are the radius, away from another point, and that other point would be the center of the circle.

So, how do we find out the distance between these two points? Between these two points? So the length of that orange line, well, we can use the distance formula, which is essentially the Pythagorean theorem. The distance squared—so if the length of that is the distance, we could say the distance squared is going to be equal to our change in x squared. So that right there is our change in x.

I don't have to write really small, but that's our change in x, plus our change in y squared. Our change in y squared. Now, what is our change in x? Our change in x—and you could even eyeball it here—looks like it's one, but let's verify it. We could view this point as the— it doesn't matter which one you view as the start or the end, as long as you're consistent.

So let's see if we view this as the end. We'd say: -5 minus -4. So this would be equal to -1. So when you go from the center to this outer point (-5, 9), you go one back in the x-direction. Now, the actual distance would just be the absolute value of that, but it doesn't matter that this is a negative because we're about to square it, and so that negative sign will go away.

Now what is our change in y? Our change in y—well, if this is the finishing y, -9 minus -7—our initial y is equal to -2. Notice just to go from that y to that y, we go to -2. So actually, we could call the length of that side as the absolute value of our change in y, and we could view this as the absolute value of our change in x. It doesn't really matter because once we square them, the negatives go away.

So our distance squared, or our distance squared—I really could call this the radius squared—is going to be equal to our change in x squared. Well, it's -1 squared, which is just going to be 1 plus our change in y squared. -2 squared is just plus 4. 1 + 4, and so you have your distance squared is equal to 5, or that the distance is equal to the square root of 5.

I could have just called this variable the radius, so we could say the radius is equal to the square root of 5, and we're done.