Functions
AI Assistant

Circle Equations

Graph circles and find their center and radius

The standard equation of a circle is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. The graph shows x² + y² = 9 — a circle centered at the origin with radius 3.

Every point on the circle is exactly r units from the center. This comes directly from the distance formula, which is the Pythagorean theorem in disguise.

Ask the AI "Shift the center to (2, 3)" or "What circle passes through (5, 0)?"

Graph

FAQ

What is the standard form of a circle equation?
The standard form is (x − h)² + (y − k)² = r². The center is at (h, k) and the radius is r. For example, (x − 2)² + (y + 3)² = 16 has center (2, −3) and radius 4.
How do I find the center and radius from an equation?
If the equation is in standard form, read them directly. If it's expanded like x² + y² − 4x + 6y − 3 = 0, complete the square for both x and y to get standard form: (x − 2)² + (y + 3)² = 16.
Why is the circle equation related to the distance formula?
The equation (x − h)² + (y − k)² = r² says "the distance from (x, y) to (h, k) equals r." That's exactly the distance formula set equal to r. A circle is just all points at a fixed distance from the center.
How is a circle different from an ellipse?
A circle has equal "stretching" in all directions — the same radius everywhere. An ellipse has two different radii (semi-major and semi-minor axes). A circle is a special case of an ellipse where both axes are equal.

Related Topics

The EllipseConic SectionsPythagorean Theorem