Standard Form
(x - h)² + (y - k)² = r²
Center: (h, k), Radius: r
General Form
x² + y² + Dx + Ey + F = 0
D = -2h, E = -2k, F = h² + k² - r²
Standard → General: Expand the squares and collect like terms.
General → Standard: Complete the square for both x and y terms.
A circle equation is a mathematical representation of all points that are equidistant from a central point. The standard form of a circle equation, (x - h)² + (y - k)² = r², directly shows the center (h, k) and radius r of the circle. The general form, x² + y² + Dx + Ey + F = 0, is an expanded version that can be converted back to standard form through completing the square.
Circle equations are fundamental in analytic geometry and have applications in physics, engineering, computer graphics, and many other fields. Understanding how to derive and manipulate circle equations is essential for solving problems involving circular paths, orbits, and geometric constructions.
- When using three points, they must not be collinear (on the same line)
- The radius must always be a positive value
- For the general form, the equation must satisfy the condition D² + E² - 4F > 0
- Results may have rounding for irrational values
Disclaimer: Circle equations are calculated using standard mathematical formulas. Results depend on correct input values and non-collinear points.