# Circle

Circle Dimensions1
ConnectedYes
CompactYes
Euler characteristic0
OrientableYes
SymmetrySO(1)
Disk Rank2
SpaceSpherical
Notation
Tapertopic notation2
Toratopic notation(II)
Bracket notation(II)
Elements
Edges1 circle
Area$\pi{}r^2$ Heightdiameter: $2r$ Central density1
Related polytopes
DualDisk
ConjugateNone
Abstract & topological properties
Euler characteristic1
OrientableYes
Properties
SymmetryO(2)
ConvexYes
NatureTame

A circle, also known as a 1-sphere is the set of all points on a 2D plane that are a certain distance away from a given point. This distance is called the radius. Circles are not considered to be polygons, as they cannot be meaningfully assigned any elements other than points.

A circle is the 1-dimensional hypersphere.

Formally, a filled-in circle is called a disk, and its boundary is called a circle. A disk is the 2-dimensional hyperball.

## Coordinates

The points on a circle are all points $(x,y)$ such that

• $x^2+y^2=r^2$ where r is the radius of the circle.

## Special Properties

Circles have many unique properties among the 2D shapes. These include:

• They are rotationally symmetric from every angle.
• They have the lowest perimeter-to-area ratio of any closed 2D shape.
• Their center of mass's cycloid is a straight line.