1. What is the Equation of a Circle?

The standard equation of a circle with center at (h, k) and radius r is (x - h)² + (y - k)² = r². This equation describes all points (x, y) that are exactly r units away from the center point (h, k).

2. How Does the Calculator Work?

The calculator uses the standard circle equation:

Where:

• \( (h, k) \) — Center coordinates of the circle
• \( r \) — Radius of the circle (must be positive)
• \( (x, y) \) — Any point on the circumference

Explanation: The equation represents the set of all points in a plane that are at a given distance (the radius) from a given point (the center).

3. Importance of Circle Equations

Details: Circle equations are fundamental in geometry, physics, engineering, and computer graphics. They're used in problems involving circular motion, wave propagation, and geometric design.

4. Using the Calculator

Tips: Enter the x and y coordinates of the circle's center and its radius. The radius must be a positive number. The calculator will output the standard form equation of the circle.

5. Frequently Asked Questions (FAQ)

Q1: What if my center coordinates are negative?
A: The calculator handles negative coordinates correctly, adjusting the signs in the equation automatically.

Q2: Can I use decimal values for coordinates and radius?
A: Yes, the calculator accepts decimal values for all inputs.

Q3: What happens if I enter zero for the radius?
A: The radius must be greater than zero. A circle cannot have a zero or negative radius.

Q4: How is this different from the general form of a circle equation?
A: The standard form directly shows the center and radius, while the general form (x² + y² + Dx + Ey + F = 0) requires completing the square to find these properties.

Q5: Can this calculator find the equation from three points?
A: No, this calculator requires the center and radius. For three-point calculations, you would need a different method involving perpendicular bisectors.