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 ${\displaystyle \mathbb R^{2}}$ is

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

here the symbol ${\displaystyle S^1}$ is the modern name used in maths and science. In contrast an open disk -in the euclidean plane- is defined as

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

as far the closed disk is

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

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

In all formulas, ${\displaystyle r}$ is the radius, ${\displaystyle d}$ is the diameter, and ${\displaystyle c}$ is the circumference.

See also

• Radius
• Circumference
• Semicircle
• Circumscribed circle
• Inscribed circle