General Form Equation Circle Calculator

Background.

The centre-radius equation of a circle is the one people find easiest to picture, but much of the algebra done with circles happens in the expanded, general form x² + y² + Dx + Ey + F = 0. Once the brackets are multiplied out, the equation is a polynomial like any other: it can be added to or subtracted from another equation, substituted into, and solved alongside a line or a second curve without carrying squared binomials around.

That is why intersection problems are usually set up in general form. To find where a circle meets the x-axis, put y = 0 and what remains, x² + Dx + F = 0, is an ordinary quadratic. To find where two circles cross, subtract one general-form equation from the other: the x² and y² terms cancel, leaving the equation of the straight line through both crossing points. Coordinate-geometry courses and the simplest algebraic methods for fitting a circle to measured points both lean on this form.

Each coefficient carries a piece of geometry. D and E are simply −2h and −2k, so they record the centre with the sign flipped and doubled. F = h² + k² − r² is the squared distance from the origin to the centre minus the squared radius. That makes F the product of the circle's two x-intercepts, and also of its two y-intercepts, whenever those intercepts exist. A negative F means the intercepts straddle the origin, which is another way of saying the origin sits inside the circle.

The page works in one direction only, from a centre and a positive radius to D, E and F, with the x² and y² coefficients fixed at 1. It does not fit a circle through measured points or handle a radius of zero.

What is general form equation circle calculator?

The general form of a circle's equation is x² + y² + Dx + Ey + F = 0, the expanded version of (x − h)² + (y − k)² = r². It belongs to the wider family of second-degree equations Ax² + Bxy + Cy² + Dx + Ey + F = 0; a circle is the case with no xy term and equal x² and y² coefficients, here scaled to 1. The three numbers D, E and F identify the circle completely.

How to use this calculator.

1. 01Enter the x-coordinate of the centre as h and the y-coordinate as k, keeping their signs. A centre written as (y + 3) in a bracketed equation has k = −3.
2. 02Enter the radius r, which must be positive. If you only know the diameter, halve it; if you know the area, r is the square root of area ÷ π.
3. 03Read D and E, then the constant F, and assemble the equation as x² + y² + Dx + Ey + F = 0, writing a negative coefficient with a minus sign in place of the plus.
4. 04To check the result, substitute any point you know is on the circle, such as (h + r, k); the left-hand side should come to zero.

The formula.

Expanding (x − h)² + (y − k)² = r² gives x² − 2hx + h² + y² − 2ky + k² − r² = 0. Grouping like terms, the x coefficient is D = −2h, the y coefficient is E = −2k, and everything without a variable collects into F = h² + k² − r². D and E depend only on the centre, so moving the circle sideways changes them while resizing it does not. F depends on all three inputs, and because r² is subtracted it drops as the circle grows. Substituting any point (x₀, y₀) into the left-hand side gives the squared distance from that point to the centre minus r²: negative inside the circle, zero on it, positive outside.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.