1. What is the Circle Equation?

The circle equation \( x^2 + y^2 = r^2 \) represents all points (x, y) that are at a fixed distance (radius r) from a center point (typically the origin). It's a fundamental equation in coordinate geometry.

2. How Does the Calculator Work?

The calculator uses the circle equation:

Rearranged to solve for y:

Where:

• \( r \) — Radius of the circle
• \( x \) — X-coordinate of a point on the circle
• \( y \) — Y-coordinate(s) corresponding to the x-value

Explanation: For any given x-value within the range [-r, r], there are two corresponding y-values (positive and negative) that satisfy the circle equation.

3. Importance of Circle Calculations

Details: Circle equations are essential in geometry, physics, engineering, and computer graphics. They help determine points on a circle's circumference, calculate areas, and solve problems involving circular motion.

4. Using the Calculator

Tips: Enter the circle's radius and an x-coordinate within the range of -r to r. The calculator will return the corresponding y-values. Note that for x-values outside this range, there are no real solutions.

5. Frequently Asked Questions (FAQ)

Q1: What if I get an error or no solution?
A: This occurs when the x-value you entered is outside the range [-r, r]. For any x-value beyond this range, there are no real y-values that satisfy the circle equation.

Q2: Can I use this for circles not centered at the origin?
A: This calculator specifically handles circles centered at the origin (0,0). For circles centered at (h,k), the equation is \((x-h)^2 + (y-k)^2 = r^2\).

Q3: What are the applications of circle equations?
A: Circle equations are used in computer graphics, navigation systems, physics (circular motion), engineering (designing circular components), and many other fields.

Q4: How accurate are the results?
A: The calculator provides results with 4 decimal places of precision, which is sufficient for most practical applications.

Q5: Can I calculate x from y instead?
A: Yes, the equation works symmetrically. You could rearrange to \( x = \pm\sqrt{r^2 - y^2} \) to find x-values from y-coordinates.