A Circle is defined as - The locus of a point which moves such that its distance from a fixed point is constant. In cartesian coordinates : x^2 + y^2 = r^2. Now obviously, a circle is a curve or a line.
What baffles me is that how can we say “area of a circle = pi()r^2”. Should’nt it be “area in a circle = pi()r^2” ? because the circle is a curve and curves have no area only length.
A circle (or any other two-dimensional shape) is also defined as the area encompassed by its periphery.
No, I don’t think the circle includes what’s inside it. However, the area of a 2-D shape is taken to mean the area that it encloses. The reason is that you never need to talk about the area of a 1-D figure, so it’s not ambiguous.
From Dictionary.com:
There are a number of ways to define a circle, and this is one of them.
Okay, fair enough. I think that’s not the most often used definition in geometry, though. If someone refers to a point on a circle, you can bet they’re talking about a point on the outside rim.
My WAG:
I’ve never seen anyone in math define a circle as anything other than the points defined by the usual x[sup]2[/sup]+y[sup]2[/sup]=r[sup]2[/sup] type definition. But of course, when Joe Average speaks of a circle, he probably means the definition that Q.E.D. gave, and circles being one of those things we’ve known about for ever and a day, the phrase “the area of a circle is [symbol]p[/symbol]r[sup]2[/sup]” has entered the language and gets used even though from a strict perspective, you’re right in saying that a circle has no area itself but only encloses an area.
It makes sense to me, anyway…
I think it’s a case of apostrophe. The full phrase to be substituted for “of” is "bounded by " .
As long as there is no ambiguity I don’t think it matters much in Math.
In my own personal life experience, I find that math and engineer types speak well, because they care about precision.
When speaking precisely, mathematicians usually use disk to refer to the interior of a circle. (A ball is the interior of a sphere. You can also refer to n-balls and n-disks to to talk about the interiors of higher-dimensional hyperspheres.)