Detailed definition
Understanding Circle
In analytic geometry, a circle is the set of all points at a fixed distance from one center. On the graph that distance is the radius, and in equation form the same idea appears through squared horizontal and vertical offsets from the center.
A circle can also be viewed as a conic section, produced when a plane cuts a cone perpendicular to its axis. In coordinate work, however, the center-radius description is usually the most practical starting point.
This page keeps the center, radius, and equation-linked graph together so you can see how a familiar geometric figure becomes an algebraic curve without losing its meaning.
Key facts
Important ideas to remember
- A circle is a conic section consisting of all points a fixed distance from one center.
- The standard center-radius equation records equal distance from the center in algebraic form.
- A circle is the special ellipse whose horizontal and vertical radii are equal.
- Changing the center shifts the graph, while changing the radius changes the size of the circle.
Where it is used
Where circle shows up
- Use the analytic circle when graphing center-radius equations or writing an equation from a graph.
- Use it in coordinate proofs involving fixed distance from a point.
- Use it as the conic model behind many geometry, physics, and design problems involving round paths.
Common mistakes
What to watch out for
- Do not confuse the center coordinates with the radius value when reading the equation.
- Do not forget that radius is a distance and must be nonnegative.
- Do not mistake an ellipse with unequal axes for a circle just because it looks round on a stretched graph.