Equation of a Circle Calculator

The standard equation of a circle is (x − h)² + (y − k)² = r², centred at (h, k).

How the Math Works

The equation of a circle is derived from the distance formula in coordinate geometry. When we know the center point (h, k) and radius r, every point (x, y) on the circle must satisfy the condition that its distance from the center equals the radius. Using the Pythagorean theorem, this distance relationship becomes (x − h)² + (y − k)² = r², where the horizontal distance (x − h) and vertical distance (y − k) form the legs of a right triangle with hypotenuse r. This standard form elegantly captures all points equidistant from the center.

Practical Applications

To use this calculator, simply enter the coordinates of the center point (h, k) and the radius r into the designated fields. The calculator will instantly display the complete circle equation in standard form, and you can also solve for specific points by substituting values for either x or y. This is particularly useful when working with circle graphs in algebra, finding intersections with lines or other circles, or verifying whether a given point lies on, inside, or outside a circle.

Day-to-Day Use

Understanding circle equations helps in everyday applications like designing circular objects, planning roundabouts or circular gardens, analyzing radar or satellite coverage areas, and even in video game development for character movement within circular boundaries. Navigation systems use circle equations to calculate distances and determine if locations fall within specific radius zones, making it valuable for delivery routes, flight path planning, and geofencing applications in smartphone apps.

Worked example

Centre (2, −3), r = 5 → (x − 2)² + (y + 3)² = 25.

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