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A circle is a geometric figure which consist of points in the Euclidean plane that are allocated at a fixed distance from the origin. This fixed distance is called the circle's radius.

The precise definition in the Euclidean plane $\R^2$ is

$S^1=\Big\{(x,y)\in\R^2:x^2+y^2=1\Big\}$

here the symbol $S^1$ is the modern name used in the math-science. In contrast an open disk -in the euclidean plane- is defined as

$D^2=\Big\{(x,y)\in\R^2:x^2+y^2<1\Big\}$

as far the closed disk is

$\bar{D}^2=\Big\{(x,y)\in\R^2:x^2+y^2\le1\Big\}$

Observe that the circle is the frontier of both, not included in the open one but included in the closed.

In informal discussion people tend to confuse a circle with a disk.

Formulas

• General equation for a circle: $(x-h)^2+(y-k)^2=r^2$ (with the origin at $(h,k)$)
• Circumference of a circle: $c=2\pi r=\pi d$
• Area bounded by a circle: $\pi r^2$ or $\frac{cr}{2}$ .

In all formulas, $r$ is the radius, $d$ is the diameter, and $c$ is the circumference.