Circle Equation Calculator

About this calculator

Every circle in the coordinate plane can be written in standard form as (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. This equation comes directly from the distance formula: any point (x, y) on the circle sits exactly r units from the center, and squaring that distance relationship removes the square root to give a clean algebraic equation.

Writing a circle this way is a core skill in analytic geometry courses, but it also has real applications: architects and CAD designers use it to define circular arcs and fillets precisely, GIS and mapping software uses circle equations to define radius-based search zones or no-fly zones, and physics students use it to describe circular motion paths. The same form also generalizes directly to spheres in 3D (adding a z-term), which is why it's a stepping stone toward more advanced conic-section and surface work.

This calculator takes a center point and radius and returns the fully expanded standard-form equation, along with the circle's area (πr²) and circumference (2πr) for a complete picture of the shape.