How this instrument works
General form writes a circle as x² + y² + Dx + Ey + F = 0 — no parentheses, no obvious center or radius, just three coefficients. Standard form, (x − h)² + (y − k)² = r², is the readable version: h and k are the center's coordinates, r is the radius, plain to see. This instrument does the algebraic inverse of the site's circle-center calculator: that tool completes the square to pull (h, k, r) back out of a general-form equation, while this one expands the binomials forward, turning a center and radius you already know into the D, E, F a graphing tool, CAD program, or textbook exercise is asking for.
The expansion is short but easy to fumble by hand. Squaring out (x − h)² + (y − k)² = r² gives x² − 2hx + h² + y² − 2ky + k² = r², and moving everything to one side collects the linear terms into D = −2h and E = −2k — note the minus sign, which is the single most common source of a wrong answer, since it is tempting to write D = 2h out of habit. The constant term F = h² + k² − r² is a genuine surprise: it is exactly the power of the origin with respect to the circle, the same quantity used to test whether a point sits inside, on, or outside a circle without computing a distance directly.
F carries a built-in check because of that fact. A negative F means the origin lies inside the circle, a positive F means it lies outside, and F = 0 means the circle passes through the origin exactly — a fast way to sanity-check an answer before trusting it. General form also has a quiet restriction worth knowing: the coefficients on x² and y² must both equal 1 and there can be no xy term, or the shape stops being a circle at all and becomes some other conic section instead.
- Enter the circle's center coordinates into Center: h and Center: k.
- Enter the circle's Radius.
- Read the results: D, E, and F, in x² + y² + Dx + Ey + F = 0, fill in automatically.
- Check your work by expanding (x − h)² + (y − k)² = r² by hand for small whole numbers and matching every term.
- If F comes out at 0, the circle passes through the origin — a quick way to confirm the center and radius you typed are the ones you meant.
Worked example — center (2, 3), radius 4
Take a circle centered at (2, 3) with radius 4 — inputs h = 2, k = 3, r = 4. Standard form reads (x − 2)² + (y − 3)² = 16. Squaring out both binomials gives x² − 4x + 4 + y² − 6y + 9 = 16, and moving the 16 to the left collects into x² + y² − 4x − 6y − 3 = 0. Matching terms against x² + y² + Dx + Ey + F = 0 gives D = −4, E = −6, and F = −3 — exactly what the sheet returns, and exactly what D = −2(2), E = −2(3), and F = 4 + 9 − 16 confirm through the direct formulas.
F = −3 is also the power of the origin with respect to this circle: the center sits √13 ≈ 3.606 units from the origin, and 3.606² − 4² works out to 13 − 16 = −3. A negative value means the origin lies inside the circle, which checks out — a radius-4 circle centered at (2, 3) easily contains the point (0, 0). A graphing calculator or CAD tool that only accepts x² + y² + Dx + Ey + F = 0 as input can now take −4, −6, and −3 directly, with no parentheses left to expand.
Questions
What is the general form of a circle's equation?
It is x² + y² + Dx + Ey + F = 0, the expanded version of standard form (x − h)² + (y − k)² = r². The coefficients relate to the center and radius by D = −2h, E = −2k, and F = h² + k² − r². Most equation solvers, CAD programs, and conic-section exercises expect this expanded shape rather than the center-radius form.
How is general form different from standard form?
Standard form shows the center and radius in plain sight; general form hides them inside three coefficients after the binomials are expanded and like terms collected. This calculator performs that expansion. The site's circle-center calculator does the opposite: it completes the square on D, E, and F to recover h, k, and r again.
Why is the constant term F more complicated than D and E?
D and E only encode the center, doubled and negated, so they stay simple. F absorbs both the center and the radius at once — it equals h² + k² − r², which happens to be the power of the origin with respect to the circle. A negative F means the origin sits inside the circle, a positive F means outside, and F = 0 means the circle passes through the origin.
What is the most common mistake when converting to general form?
Dropping the minus sign. Because D = −2h and E = −2k, a center with positive coordinates produces negative coefficients — a circle centered at (2, 3) gives D = −4 and E = −6, not +4 and +6. Writing the sign the wrong way is the single most frequent error students make expanding (x − h)² + (y − k)² = r² by hand.
Do the coefficients of x² and y² have to be equal?
Yes — for the equation to describe a circle rather than some other conic section, x² and y² must both carry the coefficient 1, and there can be no xy cross term. If those coefficients differ, or a cross term is present, the general quadratic Ax² + Bxy + Cy² + Dx + Ey + F = 0 describes an ellipse, parabola, or hyperbola instead.
How do I recover the center and radius from D, E, and F?
Reverse the formulas: h = −D⁄2, k = −E⁄2, and r = √(h² + k² − F). This is completing the square worked backwards, and it is exactly the calculation the site's circle-center calculator performs when it is given D, E, and F as its starting point.