What is the equation of a circle with center (h, k) and radius r?

The equation of a circle is derived from the definition of a circle as the set of all points that are a fixed distance (the radius) from a center point. For a circle with center coordinates at (h, k) and radius r, the formula captures this relationship perfectly.

When expressing the equation, (x - h)² + (y - k)² = r² is used. In this equation:

• (x - h) represents the horizontal distance from any point on the circle to the center's x-coordinate, h.

• (y - k) represents the vertical distance from any point on the circle to the center's y-coordinate, k.

• The squared terms ensure that both distances contribute positively to the radius measurement regardless of whether the point is to the left/right or above/below the center.

The right side of the equation, r², represents the square of the radius, establishing that the sum of these squared distances must equal the square of the radius for any point (x, y) on the circle to maintain a consistent distance from the center.

This formula is central to understanding circular geometry in the Cartesian coordinate system, as it provides a clear and concise way to identify all points that conform to the circle's definition