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Figure
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.

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    $\begingroup$ This question doesn't make any sense. $\endgroup$
    – user111187
    Sep 27 '14 at 9:36
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    $\begingroup$ Why not, @user111187 ? I think it does and it is a rather good question. $\endgroup$
    – Timbuc
    Sep 27 '14 at 9:38
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    $\begingroup$ the first is a circle and is a curve and the second is a called a disc and is a region with area $\endgroup$
    – Semsem
    Sep 27 '14 at 9:39
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    $\begingroup$ you can't show that the second figure is a circle. $\endgroup$
    – Semsem
    Sep 27 '14 at 10:01
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    $\begingroup$ @user111187: I was the person who edited it and I stand against what you say. All I have done is added more spacing and inserted articles like 'the'. One should be able to understand the question even without my edit. Just because it's not perfect english doesn't mean it didn't convey the idea. Please do not push away questions on this site without at least trying to understand what it asks. $\endgroup$
    – Nick
    Sep 27 '14 at 15:20
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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

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    $\begingroup$ Yes this is a good explanation. Thanks $\endgroup$
    – Sara
    Sep 27 '14 at 15:08
  • $\begingroup$ @Sara: Is there anything you wish for me to elaborate on? I admit I may have not been as clear as I could. $\endgroup$
    – Nick
    Sep 27 '14 at 15:22
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    $\begingroup$ FWIW, this is why mathematicians like to use mathematical notation: the difference in meaning of the sets $$C = \{ \boldsymbol x \in \mathbb R^2 : |\boldsymbol x - \boldsymbol x_0| = r\}$$ and $$D = \{ \boldsymbol x \in \mathbb R^2 : |\boldsymbol x - \boldsymbol x_0| \le r\}$$ are unambiguous. $\endgroup$
    – heropup
    Sep 27 '14 at 16:03
  • $\begingroup$ @heropup: Yes, the difference is clear but why do you recommend I use it when stating it in any other way is just as good? $\endgroup$
    – Nick
    Sep 27 '14 at 16:40
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    $\begingroup$ No, I'm not recommending you use it--I'm just commenting on the general notion that colloquial language has a tendency to be mathematically imprecise: many of us, by now, have encountered situations where trying to precisely describe a mathematical concept in English (or some other language) is surprisingly difficult or verbose. $\endgroup$
    – heropup
    Sep 27 '14 at 16:46
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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.

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  • $\begingroup$ Which means if I teach to someone about circle I can't show anything like figure 2. Which is not a circle. $\endgroup$
    – Sara
    Sep 27 '14 at 9:44
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    $\begingroup$ That's how I would do it. Of course it could be confusing when you talk about the area of a circle. Just specify that you really mean the area of the region bounded by the circle (the disc). $\endgroup$ Sep 27 '14 at 9:48
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    $\begingroup$ This is fine when a distinction is required, and becomes important when we get into more advanced work. But the formulae I learned at school included the circumference and area of a circle (not the area of a disc) - and we'd describe wheels and other similar solid objects as circular. Note also that there is no similarly convenient language for a sphere and its surface. $\endgroup$ Sep 27 '14 at 9:53
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    $\begingroup$ The sphere is the surface. The region it encloses is a ball. $\endgroup$ Sep 27 '14 at 10:00
  • $\begingroup$ When you teach people about a specialized field that uses words that are in everyday use, you sometimes - maybe often - have to start out by warning that the vocabulary is a little different from what your students might be used to. Even "lines" in geometry aren't like "lines" in everyday life - the theoretical lines extend to infinity and the "pen width" is zero. The things we draw on paper to represent "lines" and "circles" are merely approximations. $\endgroup$ Sep 28 '14 at 0:17
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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.

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    $\begingroup$ You're right, we often call the area of a shape as shape. $\endgroup$
    – Nick
    Sep 27 '14 at 16:34

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