Definition of Circle

Points at a Fixed Distance from the Center

A circle is the collection of all points on a plane that have the same distance from one fixed point. This fixed point is called the center of the circle, while the equal distance from the center to every point on the circle is called the radius.

Imagine you tie a string to a nail, then pull the string tight and draw a complete curve around the nail. The curve that forms is what we call a circle, the nail is its center, and the length of the string is its radius.

In the visualization, $P$ is the center, while $A$, $B$, and $C$ lie on the circle. The three distances $PA$, $PB$, and $PC$ are all equal to $r$. That shared distance is exactly what places the three points on one circle.

Circle as Points at a Fixed Distance

We can now state the idea precisely. A circle with center $P(a, b)$ and radius $r$ is the set of all points $(x, y)$ that satisfy:

Here, $d(P, (x,y))$ is the distance from the center $P$ to the point $(x, y)$.

If we use the distance formula in the Cartesian coordinate system, we can write it like this:

Circle Equation

From the mathematical definition above, we can derive the circle equation by squaring both sides:

This is the standard equation of a circle with center $(a, b)$ and radius $r$. For a point $(x,y)$, compare $(x-a)^2+(y-b)^2$ with $r^2$. If the first value is smaller, the point is inside the circle. Equal values put it on the circle, while a larger first value puts it outside.

For the circle in the visualization above, its equation is:

Special Form of Circle Equation

When the center is the origin $(0, 0)$, both shift terms disappear and the equation becomes:

This shorter form is useful whenever the center is already known to be the origin.

Basic Elements of a Circle

Three geometric elements appear throughout circle problems:

1. Center of circle is the fixed reference point for all points on the circle. Every point on the circle has the same distance to this center.

2. Radius is the distance from the center of the circle to any point on the circle. In one circle, all radii have the same length.

3. Diameter is a line segment that connects two points on the circle and passes through the center. Its length is twice the radius, or $d = 2r$.

The visualization shows how the radius and diameter share the same center:

Finding an Equation from Center and Radius

Substitute the given center and radius into the standard form.

Example: Determine the equation of a circle centered at $(3, -2)$ with radius $5$.

Solution: Substitute the given center and radius into the standard equation:

So the circle equation is $(x-3)^2 + (y+2)^2 = 25$.