What Are Circles, Geometrically?

The Circle

The definition of a circle is the locus of points that are a fixed distance $r$ from a single point $P$. But it really isn't that complicated! Draw a point $P$ and draw many segments with length $r$ and $P$ as one endpoint. The set of all such points $A$ looks like this round thing:

The standard equation for a circle in the coordinate plane will include the center point $P$ and the radius $r$. We can also define a circle by two fixed points (one as the center, the other as a point on the circle) or three points on the circle.

The circle has many nice properties regarding its symmetry.

Circles are made of points at a constant distance from the center

A circle has a center and a radius, $r$. It's made of the points at a constant distance $r$ away from the center.

Circles appear everywhere in nature and math, not just for their simplicity, but also for their remarkable symmetries (both rotational and reflectional), and for their close connection to angles. It's no coincidence that wheels and protractors both come in the shape of a circle! And since that shape is governed by the famous constant pi, you can bet that pi (π) appears everywhere too.

The image above depicts a blue circle, with center O and radius OA drawn in red. Points B and C are also on the circle, so OB and OC are also radii (segments of the same length as OA), just not drawn in the image.