1. What is the Circle Equation?

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

2. How Does the Calculator Work?

The calculator uses the circle equation:

Where:

• \( (h, k) \) — Center coordinates of the circle
• \( r \) — Radius of the circle (must be positive)
• \( (x, y) \) — Coordinates of a point to test

Explanation: The calculator determines if a given point lies on the circle by substituting the values into the equation and checking if both sides are equal.

3. Importance of Circle Equation

Details: The circle equation is fundamental in geometry, physics, engineering, and computer graphics for representing circular shapes and calculating distances.

4. Using the Calculator

Tips: Enter the circle's center coordinates (h, k), radius (r), and the point coordinates (x, y) you want to test. Radius must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What if I only know the diameter?
A: The radius is half the diameter. Divide your diameter by 2 before entering it as the radius.

Q2: How accurate are the results?
A: Results are accurate to 4 decimal places. Points within 0.0001 units of the circle's edge are considered "on" the circle.

Q3: Can I use this for 3D circles (spheres)?
A: No, this is for 2D circles only. The sphere equation includes a z-coordinate: \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\).

Q4: What does a negative radius mean?
A: Radius must be positive. Negative values will be treated as invalid input.

Q5: How is the distance calculated?
A: Distance from center to point is calculated using the distance formula: \(\sqrt{(x-h)^2 + (y-k)^2}\).