A **circle** is a geometric figure which consist of points in the Euclidean plane that are allocated at a fixed distance from the circle centre. 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 maths and 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.

## See also

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