For elementary questions concerning circles (or disks). A circle is the locus of points in a plane that are at a fixed distance from a fixed point. Use this tag alongside [geometry], [Euclidean geometry], or something similar. Do not use this tag for more advanced topics, such as complex analysis or topology.
A circle is a shape in geometry, defined as the locus of points that have a fixed distance from a certain point, called the centre. The fixed distance from the centre of a circle to any of its points is called the radius. The length of the set of points is called the circumference, and for Euclidean space is related to the length of the radius by:
\begin{equation}\text{circumference}=2\pi\times\text{radius}\end{equation}
Similarly, in Euclidean space the area enclosed by a circle is given by:
\begin{equation}\text{area}=\pi\times\text{radius}^2\end{equation}
Because of their radial symmetry and structure, circles have a large number of desirable properties. These include:
- The circle is the shape with the largest area for a given length of perimeter.
- The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle.
- All circles are similar.
- A circle's circumference and radius are proportional.
- The area enclosed and the square of its radius are proportional.
- The circle that is centred at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the unit sphere, it becomes the Riemannian circle. Through any three points, not all on the same line, there lies a unique circle.
- In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points.
There are many more properties of circles, see the following source for more information: https://en.wikipedia.org/wiki/Circle
