Circle Equation
A circle is the set of all points in a plane at a given distance, called the radius, from a given point called the center. The equation of a circle is a way to express the definition of a circle on the coordinate plane. When the center of the circle is at the point `C(x_1,y_1)`, the equation becomes `(x - x_1)^2 + (y - y_1)^2 = r^2`, where `r` is the radius.
A circle with the equation `x^2 + y^2 = 36` is a circle with its center at the origin and a radius of 6.
Created with GeoGebra by Vitor Nunes
Interactivity
Try moving the point `C` (which defines the center of the circle) and change the value of `r`.
The circle (of brown color) defines the points of the plane that are all at the same distance from the center. This distance is `r` (the radius of the circle).
Circle on a Graph
Circle equations are often found in the general form of `ax^2 + by^2 + cx + dy + e = 0`. Unfortunately, it can be difficult to decipher any meaningful properties about a given circle from its general equation. Using the Completing the Square technique converts the equation to an easier form, which contains values for the center and radius of the circle.
