Equation for a Circle

The standard equation for a circle centered at the origin is x² + y² = r², where r represents the radius. I’ve decided to include this concept into the unit because it turns out that the standard formula of a circle is derived from the distance formula you learned earlier. Take, for example, the circle centered at the origin below:

We don’t know the coordinates of point P, but if we use try to find its length using the formula for distance, we’ll end up with (x² – y²) = distance, where distance represents the radius. Subsequently, squaring both sides gives us x² + y² = r². Recognize?

In the video below, question 1 asks us to write in standard form a circle whose center is at the origin, while question 2 has a circle whose center isn’t centered at the origin. Notice how the formula differs to accommodate that change.

Finally, the last thing you need to be aware of is how to verify if a point lies within a circle for a given radius. This next video will help you make that connection.


a) Determine an equation and the radius for the circle that has its center at the origin and passes through the point A(6,−8).

b) Is the point A(−5,9) inside this circle?