How this calculator works
A circle can be written in general form as x² + y² + Dx + Ey + F = 0, or in centre-radius form as (x - h)² + (y - k)² = r². The centre-radius form reads the centre (h, k) and radius r directly. To convert general form, complete the square for x and for y. The terms Dx and Ey split into a perfect square plus a constant, which moves to the other side and becomes the radius squared. The centre is (-D/2, -E/2) and the radius is the square root of (D² + E²)/4 - F. If that quantity is negative, no real circle exists. The point test compares the distance from the centre with the radius: equal means on the circle, shorter means inside, longer means outside.
Formula
General form x² + y² + Dx + Ey + F = 0 has centre (-D/2, -E/2) and radius r = sqrt((D² + E²)/4 - F). Centre-radius form (x - h)² + (y - k)² = r² expands to x² + y² - 2hx - 2ky + (h² + k² - r²) = 0, so D = -2h, E = -2k and F = h² + k² - r².
Worked example
The general equation x² + y² - 6x - 8y + 21 = 0 has D = -6, E = -8 and F = 21. The centre is (3, 4). The radius squared is 9 + 16 - 21 = 4, so the radius is 2 and the standard equation is (x - 3)² + (y - 4)² = 4. The point (0, 0) is 5 units from the centre, which is more than the radius, so it lies outside the circle.
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Frequently asked questions
Why is the centre (-D/2, -E/2) rather than (D/2, E/2)?
Completing the square turns Dx into (x + D/2)² - D²/4. The centre-radius form has (x - h), so h must equal -D/2. The same sign change applies to the y coordinate.
What does a negative radius squared mean?
It means the equation has no real points. The left side would have to equal a negative number, which the sum of two squares cannot do. Check the coefficients or the chosen F value.
Can the radius be zero?
Yes. A radius of zero describes a single point at the centre. The equation still has the correct centre and coefficients, but the circle has collapsed to one point.
How do I use the point test?
Enter the centre and radius, then the point coordinates. The calculator finds the distance from the centre. A distance equal to the radius places the point on the circle; a shorter distance places it inside; a longer distance places it outside.