What does the Circle really mean?

Question

Which of the following figure is really the circle? If a point is on the circle it means that point should be on the circumference is it? (point $Z$ on figure 1).

Point $P$ on figure $1$ is inside the circle not on the circle.
Which means the circle is like a ring with no inside.

In figure $2$, the circle is cut from a paper and hence has an inside.

There are two points $Q$ and $X$.

$X$ is on the circumference so we can say it is on the circle. Now what a about the point $Q$ can we say it is on the circle?

I am confused with the definition of circles.

Answer 1

The circle is defined as the locus of a point equidistant from a fixed point.

If we say the center is at $(a,b)$ then the locus of a point $(x,y)$ at a distance $r$ creates a curve (a boundary of length $2\pi r$) which bounds an inner area of $\pi r^2$ Then, this circle, on the Cartesian plane can be represented by the equation: $$(x-a)^2 + (y-b)^2 = r^2$$

Figure $1$ shows a circle. It can be noted that point $Z$ satisfies the given equation while point $P$ does not satisfy. Hence, we say $Z$ lies on the circle.
Note: It is not wrong to say that $P$ lies in the circle.

Now, what is Figure $2$? It is a disk (also spelled disc). A disk is defined as the region in a plane bounded by a circle. Both $X$ and $Q$ clearly lie on the disk.

But there is a problem when thinking about figure $2$. It is clearly intended that $X$ lies on the periphery of the darkened area. But does this mean that $X$ lies on the circle corresponding to the disk?

I express this as a problem because the area inside the circle defined earlier can be denoted as: $$(x-a)^2 + (y-b)^2 \lt r^2$$

Now, point $Q$ satisfies this but it is unclear whether $X$ satisfies it.

But don't be confused, remember the remark I made earlier. "Both $X$ and $Q$ clearly lie on the disk." This is because a disk is conventionally taken to be a closed disk (unless mentioned otherwise to be open). That is, the disk is defined as the set of all points which satisfy: $$ (x-a)^2 + (y-b)^2 \le r^2$$

Hopefully, my answer was helpful :D

Answer 2

Usually a circle is defined to be the set of all points $(x,y)$ in the plane that are a fixed distance from some point $p=(p_1,p_2)$ in the plane. For instance $9=(x-p_1)^2+(y-p_2)^2$ describes the circle centered at $p$ with radius 3 (by the Pythagorean Theorem).

The solid region is usually referred to as a disc, the set of all points in the plane that have distance from $p$ less than or equal to a given number.

Answer 3

I think the confusion arises from ambiguity of the word "circle" as in used as a mathematical term and as an ordinary word. As a mathematical term, circle only means Figure 1, but as a term in ordinary life, it can mean either Figure 1 or Figure 2.