Standard to General Form Circle - Center (h, k) to D, E, F

A standard to general form circle calculator turns a circle written as (x - h)^2 + (y - k)^2 = C into the expanded form x^2 + y^2 + Dx + Ey + F = 0, where D = -2h, E = -2k, and F = h^2 + k^2 - C. The two forms describe the same circle; the standard form keeps the center and radius visible, and the general form is the expanded form.

• • Expand a textbook standard-form equation: Type the h, k, and C from the standard form and copy the matching x^2 + y^2 + Dx + Ey + F = 0 into your answer.
• • Convert for a graph or sketch: Use the general form coefficients with graphing software that expects the x^2 + y^2 + Dx + Ey + F = 0 input.
• • Get D, E, and F faster: Skip expanding (x - h)^2 + (y - k)^2 = C by hand; the calculator outputs the same D, E, F.
• • Check a worked example: Verify the coefficients from a textbook example.

The standard to general form circle calculator treats the standard form as the input and the general form as the output. The standard form is the form most algebra and precalculus problems hand you first because the center and radius are visible at a glance. The general form is the same circle with the squares expanded.

If the problem runs the other way, the Circle Equation Calculator recovers the center and radius from the expanded equation.

The calculator reads the center h, k and the radius squared C from the standard form, then applies three identities to write out the general form. Results update as you type.

• h, k: Coordinates of the circle's center in the standard form (x - h)^2 + (y - k)^2 = C.
• C: Right-hand side of the standard form, equal to r^2. Must be positive for a real circle.
• D: Coefficient of x in the general form. Equal to -2h.
• E: Coefficient of y in the general form. Equal to -2k.
• F: Constant term. Equal to h^2 + k^2 - C.

The three identities fall out of one piece of algebra. Expand (x - h)^2 + (y - k)^2 = C, move C to the left, and group the x and y terms to recover x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - C) = 0.

According to Math Open Reference, the standard equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) are the coordinates of the center and r is the radius.

According to Cuemath, expanding (x - h)^2 + (y - k)^2 = r^2 gives the general form x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0, with D = -2h, E = -2k, and F = h^2 + k^2 - r^2.

When you have both forms and want every measurement at once, the Standard Equation Circle Calculator handles the standard form from a center and one extra piece of information, and pairs with the conversion here.

Four ideas show up every time you convert a circle from standard to general form. Knowing them turns the calculator from a black box into something you can check by hand.

The same four pieces of information describe the circle in either form. h and k become the linear coefficients D and E, and C combines with the center in F. The radius r is sqrt(C).

If the problem starts from a known center and a radius, a point on the circle, or the diameter endpoints, the Circle Diameter Calculator writes the standard form for you before you run the conversion.

The standard to general form circle calculator is straightforward: read h, k, and C off the standard form, type them in, and read the D, E, F coefficients and the full general form from the results panel. Results update on every keystroke.

1. 1 Identify the standard form: Make sure the equation is in the form (x - h)^2 + (y - k)^2 = C.
2. 2 Read off h, k, and C: h is the number after x inside the first squared binomial, k is the number after y inside the second, and C is the right-hand side. Watch the sign: (x - 3)^2 means h = 3, (x + 2)^2 means h = -2.
3. 3 Type the three values: Enter h in the first field, k in the second, and C in the third. The results panel updates as you type.
4. 4 Read the D, E, F values: D, E, and F are the three coefficients that go in front of x, y, and the constant in the general form.
5. 5 Copy the full general form: The general form field writes the entire equation x^2 + y^2 + Dx + Ey + F = 0 in one line, ready to paste.
6. 6 Check the no-real-circle flag: If C is zero or negative, the page flags the result so the radius and area do not show a misleading positive number.

A textbook asks for the general form of (x - 3)^2 + (y - 4)^2 = 25. Type h = 3, k = 4, C = 25. The standard to general form circle calculator returns D = -6, E = -8, F = 0, and the general form x^2 + y^2 - 6x - 8y = 0.

When you only know three points on the circle and need the center h, k, the Circle Center Calculator solves that workflow and feeds straight into the standard form here.

What the calculator returns and how each output pays off in an algebra or coordinate-geometry task.

• • Skip the expansion by hand: Recover D, E, and F from (x - h)^2 + (y - k)^2 = C without expanding the squared binomials.
• • Avoid sign errors: The standard form uses (x - h), so a positive h shifts the circle right. The calculator applies the sign convention in D = -2h.
• • Get every measurement in one pass: The radius, diameter, area, and circumference fall out of C alongside the D, E, F coefficients.
• • Built-in no-real-circle flag: If C is zero or negative, the page flags the result instead of returning a fake radius from sqrt of a non-positive number.
• • Ready-to-paste equation text: The general form field is written as a single copyable string, ready to paste into homework or graphing tools.

These benefits stack in the typical workflow. A student reading 'convert to general form' gets the exact equation to hand in. A teacher or tutor checking a peer's answer gets the same coefficients and a radius to sanity-check the result.

If you already have the radius in hand and only need the area and circumference, the Circle Calculator skips the equation step entirely.

A few characteristics of the input change the result, the sign of the coefficients, or whether the page flags the equation as a real circle.

• • Inputs are treated as plain numbers, so the calculator does not parse symbolic values such as 'sqrt(2)' or '5 pi'. Convert those by hand first.
• • Results are decimals, not exact multiples of pi. If the answer needs to be in terms of pi, keep C = r^2: area = pi C and circumference = 2 pi sqrt(C).

The algebra is exact, so the only place rounding enters is in the decimal display.

As published by Wolfram MathWorld, a circle of radius r has area pi r^2 and circumference 2 pi r, which is why r = sqrt(C) is the bridge to every other measurement on this page.

When the same circle is described by a different formula such as parametric or polar form, the Circle Formula covers the formulas on the other side of the conversion.