General Form Equation Circle Calculator - Center, Radius, Standard Form

A general form equation circle calculator reads D, E, and F from x^2 + y^2 + Dx + Ey + F = 0 and returns the center (h, k), radius, area, circumference, and matching standard form.

• • Read center and radius from a textbook problem: Most algebra questions start with x^2 + y^2 + Dx + Ey + F = 0. Type the three coefficients and read both without completing the square by hand.
• • Convert to the standard form: Use the recovered h, k, and r^2 to write (x - h)^2 + (y - k)^2 = r^2.
• • Pull area and circumference from the same input: Once r is known, area pi r^2 and circumference 2 pi r are right there, so a sketch can be sized without retyping numbers.
• • Check whether the equation describes a real circle: If r^2 is zero or negative, the page flags the input as 'No real circle'.

The general form is what most algebra problems hand you first. The x^2 and y^2 coefficients are both 1, and D, E, F encode the center and the radius.

This page accepts the general form as input and returns every measurement a coordinate-geometry problem can ask for. When the equation is given in standard form (x - h)^2 + (y - k)^2 = r^2 instead, the reverse workflow lives on the standard to general form circle page.

When the problem runs the other direction and starts from (x - h)^2 + (y - k)^2 = r^2, Standard to General Form Circle Calculator expands it into the same x^2 + y^2 + Dx + Ey + F = 0 this page reads.

The calculator reads D, E, and F from x^2 + y^2 + Dx + Ey + F = 0, applies three identities to recover the center and radius squared, and renders the standard form and every measurement in plain text.

• D: Coefficient of x in x^2 + y^2 + Dx + Ey + F = 0. Sets h = -D / 2.
• E: Coefficient of y. Sets k = -E / 2.
• F: Constant term. Used in r^2 = (D / 2)^2 + (E / 2)^2 - F.
• r^2: Squared radius. Must be positive for a real circle.
• (h, k): Center, recovered from -D / 2 and -E / 2.

Expanding (x - h)^2 + (y - k)^2 = r^2 gives x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0. Matching coefficients with the general form yields D = -2h, E = -2k, and F = h^2 + k^2 - r^2, which solve to h = -D / 2, k = -E / 2, and r^2 = (D / 2)^2 + (E / 2)^2 - F.

Once r^2 is known, every other measurement is one line. Diameter is 2r, area is pi r^2, and circumference is 2 pi r. The page rounds to four decimal places and uses the same rounded values in the plain-text standard form.

According to Cuemath, the general form x^2 + y^2 + Dx + Ey + F = 0 expands from the standard form with D = -2h, E = -2k, and F = h^2 + k^2 - r^2.

If the equation is given in either form and the page should pick the conversion direction automatically, Circle Equation Calculator reads both forms and returns the same center, radius, area, and circumference in one panel.

Four ideas cover every step from the general form to the center, radius, and standard form.

The standard form (x - h)^2 + (y - k)^2 = r^2 is what the problem usually wants, but the general form is what the textbook gives you first. Going from the general form to the standard form by hand is completing the square twice, which is a reliable source of sign errors when the center is shifted by an odd number.

Using h = -D / 2 and k = -E / 2 collapses both completions into one step. The center is read off the linear coefficients, and r^2 is read off the constant F and the same halves.

According to Math Open Reference, the general form x^2 + y^2 + Dx + Ey + F = 0 has center (-D/2, -E/2) and radius sqrt((D/2)^2 + (E/2)^2 - F), matching the standard form exactly.

When the problem starts from three boundary points on the circle instead of an equation, Circle Center Calculator solves the (h, k) side of the same problem before the radius is even needed.

Type the three coefficients of x^2 + y^2 + Dx + Ey + F = 0 and read the center, radius, area, circumference, and standard form from the results panel.

1. 1 Read D, E, and F off the equation: D is the coefficient of x, E is the coefficient of y, and F is the constant on the left of the equals sign. Watch the signs: x^2 + y^2 + 4x - 6y - 12 = 0 means D = 4, E = -6, F = -12.
2. 2 Type the three values into the calculator: Enter D in the first field, E in the second, and F in the third. The results panel updates on every keystroke.
3. 3 Read the center (h, k): The center is shown at the top of the results panel as a single (h, k) coordinate pair. If the equation does not describe a real circle, this row shows 'No real circle'.
4. 4 Read the radius, diameter, area, and circumference: These four measurements fall out of r = sqrt(rSquared). Diameter is 2r, area is pi r^2, and circumference is 2 pi r.
5. 5 Copy the standard form: The standard form row writes (x - h)^2 + (y - k)^2 = r^2 in plain text for a homework answer or graphing tool.

A textbook gives x^2 + y^2 + 4x - 6y - 12 = 0. Type D = 4, E = -6, F = -12. The page returns center (-2, 3), radius 5, diameter 10, area 78.5398, circumference 31.4159, and standard form (x + 2)^2 + (y - 3)^2 = 25.

If you would rather enter the center h, k and a radius, or the center and one point on the circle, Standard Equation Circle Calculator is the standard-form complement to this page.

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

• • Skip completing the square: Recover the center and radius from x^2 + y^2 + Dx + Ey + F = 0 without completing the square twice by hand.
• • Avoid sign errors: The half-coefficients flip sign, so h = -D / 2 and k = -E / 2. The page applies that sign convention for you.
• • Pull every measurement at once: The center, radius, diameter, area, and circumference all fall out of D, E, F without retyping the equation.
• • Read the matching standard form: The standard form row renders (x - h)^2 + (y - k)^2 = r^2 in plain text for homework or graphing software.
• • Built-in no-real-circle flag: If r^2 is zero or negative, the page flags the input instead of returning a fake positive radius.
• • Real-time results: Every change to D, E, or F updates the results panel immediately for partial-input comparisons.

These benefits stack in a typical workflow. A student reading 'find the center and the radius' gets both quantities immediately, and a teacher checking an answer gets the same numbers plus the matching standard form.

When the radius is already known and only the area and circumference matter, Circle Calculator does the arithmetic from a single radius input without re-entering the equation.

A few characteristics of the input shape the output and the sign of the recovered center.

• • Inputs are plain numbers, so the calculator does not parse symbolic expressions such as 'sqrt(12)' or '5 pi'.
• • Results are decimal, not exact multiples of pi. Keep r^2 from the panel and remember area is pi r^2 and circumference is 2 pi r.

The algebra behind the general form identities is exact, so rounding enters only in the decimal display.

As published by Wolfram MathWorld, a circle of radius r centered at (a, b) has area pi r^2 and circumference 2 pi r.

When the same circle is described by polar coordinates or by a parametric pair (x = a + r cos t, y = b + r sin t), Circle Formula collects those formulas on the other side of the conversion.