We have seen in the tutorial on Distance Between Points how the
Pythagorean theorem can be used to obtain the distance between two points. If you wish to review that topic, click on the link indicated in blue.
Now, we are going to assume that instead of looking for the distance between two points that we have one point (2,3) and we wish to find a second point. The only requirement is that the second point have a distance of 3 units from the first point. The following diagram contains the four most obvious points that satisfy this condition.
The following diagram contains some other points and their approximate coordinates. Using
Pythagoras, we can verify that they also are 3 units away from (2,3).
| Point | Distance from (2,3) |
| (-0.6364, 4.4316) |
 |
| (4.2121, 5.0265) |
 |
| (1.0000, 0.1716) |
 |
| (3.8485, 0.6371) |
 |
With these examples, we can generalize to conclude that the set of all points that reside 3 units away from (2,3) have the following form. This set of points is called a circle with center (2,3) and radius 3.
Algebraically, for a point (x, y) to belong to the circle with center (2, 3) and radius 3, the distance from (x, y) to (2, 3) must be equal to 3 or 
. As it is more convenient to eliminate the square root, we can square both sides of the formula to obtain 
. This is the formula that those (x,y) that belong to the circle with center (2, 3) and radius 3 must satisfy.
To generalize, given a center (a, b) and a radius r, the associated circle consists of all points (x, y) that are distance r from the point (a, b).
To determine whether a point (x, y) belongs to this circle, we test that the distance from (x, y) to (a, b) 
. For convenience, we eliminate the square root by squaring both sides to obtain 
.
This is the general formula for those (x, y) that belong to a circle with center (a, b) and radius r. Practice obtaining the formulas for various radius and centers with the following interactive demonstration.
