Circle Equation Calculator

The standard form of a circle equation is:

$$(x - h)^2 + (y - k)^2 = r^2$$

where $$(h, k)$$ is the center and $$r$$ is the radius. Expanding this gives the general form:

$$x^2 + y^2 + Dx + Ey + F = 0$$

where $$D = -2h$$, $$E = -2k$$, and $$F = h^2 + k^2 - r^2$$.

The area enclosed by the circle is $$A = \pi r^2$$, and the circumference (perimeter) is $$C = 2\pi r$$.

For the point-on-circle test, the calculator computes the distance from the test point $$(p_x, p_y)$$ to the center:

$$d = \sqrt{(p_x - h)^2 + (p_y - k)^2}$$

If $$d < r$$, the point is inside the circle. If $$d = r$$ (within numerical tolerance), the point is on the circle. If $$d > r$$, the point is outside.

The standard form immediately reveals the center and radius, making it ideal for graphing. The general form is useful for algebraic manipulations, such as finding intersections with lines or other circles.

The test point position output uses codes: 1 for inside, 2 for on the circle, and 3 for outside. This is equivalent to checking the sign of $$(p_x - h)^2 + (p_y - k)^2 - r^2$$: negative means inside, zero means on, positive means outside.

The general form coefficients $$D$$, $$E$$, $$F$$ can be used to recover the center as $$(-D/2, -E/2)$$ and the radius as $$\sqrt{D^2/4 + E^2/4 - F}$$, which is the standard technique for completing the square.