Center and Radius
According to our definition, two elements define a circle: the center (a point) and the radius (a distance). If you know both the center and the radius of a circle, then you can draw the circle. Conversely, to draw a circle, you need to know both its center and its radius. It's all about give and take, isn't it?
Think about it like this: Since every circle is the same shape (uh…round), the only things that distinguish two circles are where they are and how big they are. The center tells you where the circle is. The radius tells you how big it is.
Here we have a circle with center O (written ⊙O). Point A is on the circle, making the distance OA the radius of the circle.
Unfortunately, there's a little ambiguity in the term "radius." We use it to refer to a line segment with endpoints at the center of the circle and a point on the circle as well as the length of such a segment. In the figure above, OA and OB are radii of ⊙O.
Since all radii of the circle have the same length (the radius of the circle), we can prove that any two radii of the same circle are congruent.
For instance, given that points A and B are on ⊙O, we can prove that OA and OB are congruent using the definition of a circle. The distances OA and OB are both equal to the radius of the circle because all points on the circle are equidistant from O. In other words, OA = OB. By definition of congruence of line segments, segments OA and OB are congruent.
We might call this the "wheel theorem," since it's what makes wheels work. Seriously, it's "wheely" important.
Back in the pioneer days, a homesteading family (call them the Smiths) might have driven their wagon out into the middle of a flat, wide-open plain of untamed wilderness, hammered a stake into the ground, and claimed "all land within ten miles of the stake" as their property. What shape would the Smith family's property be?
Looks like a circle, right? It even has a center (the stake) and a radius (10 miles.) According to our definition, though, a circle is only the set of points exactly a certain distance away from the center, whereas the Smiths' property includes all points within that distance. A fence enclosing the property might be considered a circle, but the Smiths own more than the fence. Otherwise there's not much point in building the fence at all, is there?
So, while the boundary of the property is indeed a circle (with radius 10 miles and center at the stake), the entire property is technically the inside of that circle—a region we call a disk. If it helps, you can think of CDs and not pioneers.
The Smiths' fence divides the world into three regions: the set of points inside the fence, the set of points outside the fence, and the set of points on the fence. Similarly, any circle divides the plane into three regions.
Given ⊙O with radius r and a point P in the same plane as ⊙O:
• P is in the interior of ⊙O if OP < r
• P is in the exterior of ⊙O if OP > r
Finally, by definition of circle, we already know that P is on ⊙O if OP = r.
Suppose that circle ⊙O has a radius of 3 inches. If the distance OP is 3 inches, is P on the exterior, interior, or on ⊙O?
We can start by comparing the distance from P to O to the radius of ⊙O. In this case, OP = 3 inches and the radius r of ⊙O is 3 inches. Since 3 = 3, we know that OP = r. By definition that means P is on ⊙O.
With a circle, we can organize and tame any wide-open plane, just as the Smiths corralled the wilderness with their stake and circular fence. And it's no coincidence that "pioneer" contains the word "pi."