We'll stick with the nice, clean, simple equation we've got here.

You'll end up with a messy plus-or-minus square root. Go ahead and try solving our implicit equation for y. (Remember those from algebra?) A formula with all the variables on one side is called implicit, while a formula that has been solved for one variable in terms of the other (such as y = mx + b) is called explicit. You may notice that we left the x and y terms on the same side of the equation, rather than solving for y as we like to do for linear equations. We now have an equation that relates the x- and y-coordinates of any point on a circle with radius r centered at the origin. Let's cover all our bases and call it r units. It could be 6, or 7, or 50, or even a million units. Pythagoras tells us that x 2 + y 2 = 5 2.īut the radius of a circle centered at the origin doesn't have to be 5 units. If ( x, y) is on the circle and the circle's center is at the origin, then the hypotenuse has a length equal to the circle's radius. The hypotenuse connects ( x, y) and the origin. The other leg is vertical and has length y. Notice that by nature of the Cartesian system, we can easily draw a right triangle based on any point ( x, y). We're trying to find an equation relating the coordinates of a generic point on the circle ( x, y). Notice that the circle goes through the points (5, 0), (0, 5), (-5, 0), and (0, -5). Let's start by considering a circle with its center at the origin and radius 5 units. But before we go there, we'll make things a little easier. We can describe a circle in the coordinate plane with an equation.

Equation of a circle