Standard Equation of a Circle Calculator
Convert a circle between center-radius, standard, parametric, and general forms, then review its key geometric properties.
Enter a circle
Accepted format: U.S.-style decimals with an optional sign and correctly grouped commas. Scientific notation and decimal commas are rejected to prevent ambiguous interpretation.
Standard equation
(x – 3)² + (y + 2)² = 25
Center
(3, – 2)
Radius
5
Radius squared
25
Diameter
10
Circumference
31.415927
Area
78.539816
x = 3 + 5 cos(α), y = – 2 + 5 sin(α)
x² + y² – 6x + 4y – 12 = 0
Standard form
(x – A)² + (y – B)² = r²
A and B locate the center. The right side is the square of the positive radius.
General conversion
D = – 2A, E = – 2B, F = A² + B² – r²
Expanding the two squared binomials produces the general equation with zero on the right.
Geometric properties
d = 2r, C = 2πr, area = πr²
All reported lengths use the same unspecified unit as the radius; area uses that unit squared.
Property details
| Property | Exact expression | Decimal value |
|---|---|---|
| Center x-coordinate | A | 3 |
| Center y-coordinate | B | – 2 |
| Radius | r | 5 |
| Radius squared | r² | 25 |
| Diameter | 2r | 10 |
| Circumference | 10π | 31.415927 |
| Area | 25π | 78.539816 |
How to use the standard equation of a circle calculator
What this calculator does
This calculator describes one real circle in several equivalent coordinate forms. It accepts either the center and radius, the coefficients in the standard equation (x – A)² + (y – B)² = C, or the coefficients in the general equation x² + y² + Dx + Ey + F = 0. From that input it identifies the center, positive radius, radius squared, diameter, circumference, area, parametric equation, and equivalent general equation. The result is an exact algebraic conversion when the input coefficients are exact; circumference and area are also shown as decimal approximations involving π. It does not graph the circle, solve intersections with other curves, or infer measurement uncertainty.
When to use it
Use the calculator when checking algebra homework, converting a completed-square equation into expanded form, recovering a center and radius from a general equation, or preparing circle parameters for analytic geometry, CAD, simulation, or programming. It is also useful for verifying whether a proposed general equation actually represents a non-degenerate real circle: the recovered radius squared must be strictly positive.
How to calculate
- The calculator opens with a ready-to-use demonstration circle centered at (3, – 2) with radius 5. Its outputs and a validated example workbook are available immediately.
- Choose Input form. Keep Center coordinates and radius, or switch to Standard-form coefficients A, B, C or General-form coefficients D, E, F. A valid current circle is converted automatically when you change forms.
- Replace the visible values. Results update live. Read the main Standard equation, then review the property cards, the two alternative equations, and the property table.
- Select Download Excel to create a fresh workbook from the current validated controls. Select Reset to clear the demonstration and all coefficient fields. Reset may disable the workbook button until a complete valid circle is entered again.
Input guide
Input form is a required selector, not a numeric quantity. It determines which three coefficients are read. Switching forms does not alter a valid circle; it translates the current center and radius into the selected representation.
Center x-coordinate and Center y-coordinate are required real numbers in the same coordinate unit. Examples are 3 and – 2. Increasing either value translates the circle horizontally or vertically without changing its size. A common mistake is to read the visible sign in a binomial literally: the term (x + 2) means A = – 2 because it is equivalent to x – ( – 2).
Radius is a required positive real length, such as 5. A larger radius increases diameter linearly, circumference linearly, and area quadratically. Zero and negative radii are rejected because this tool models a non-degenerate circle rather than a point or an imaginary locus.
Standard coefficient A and Standard coefficient B are required real numbers and equal the center coordinates. Standard coefficient C is required and must be positive because C = r²; for a radius of 5, enter 25 rather than 5. The calculator takes the positive square root of C to recover the radius.
General coefficient D, General coefficient E, and General coefficient F are required real numbers. For the startup example they are – 6, 4, and – 12. The center is ( – D/2, – E/2), while radius squared is (D² + E²)/4 – F. If this final expression is zero or negative, the equation does not describe the positive-radius real circle required by this calculator.
All numeric controls accept an optional sign, a period as the decimal separator, and properly grouped commas such as 1,250.5. Scientific notation, unit symbols, and decimal commas are rejected visibly. This strict convention prevents an entry such as 1,5 from being silently interpreted as 15.
Output guide
Standard equation is the primary exact identity. Center reports (A, B), while Radius is the positive square root of the right-hand constant. Radius squared is C and is especially useful when moving between standard and general forms. Diameter is 2r. Circumference is 2πr in linear units, and Area is πr² in square units. Larger absolute center coordinates do not affect these size properties; only the radius does.
Parametric equation gives x = A + r cos(α) and y = B + r sin(α), so varying α traces every point on the circle. General equation is the expanded polynomial with all terms on the left and zero on the right. The Property details table repeats the center, radius, radius squared, diameter, circumference, and area with both an exact expression and a decimal value. These are mathematical identities or conversions, not statistical estimates.
Worked example
Learn more
For a derivation from the distance formula and worked examples, see OpenStax's section on distance, midpoint formulas, and circles. For more detail on converting equations by completing the square, consult the Mathematics LibreTexts lesson on circles in standard and general form. OpenStax also introduces how a circle can be traced with two coordinate functions in its overview of parametric equations.
How the conversions work
The standard equation comes directly from the distance formula. Every point (x, y) on a circle is exactly r units from its center (A, B), so the squared horizontal and vertical displacements add to r². Expanding the squares creates the general coefficients D = – 2A, E = – 2B, and F = A² + B² – r².
Going in the opposite direction is completing the square. The terms x² + Dx become (x + D/2)² minus the balancing constant D²/4, and the y terms behave the same way. After those constants are moved to the other side, the right-hand value is r² = (D² + E²)/4 – F. A strictly positive result yields a real circle. A zero result collapses to a single point, while a negative result has no real points.
Common interpretation mistakes
- Confusing C in the standard equation with the radius instead of radius squared.
- Forgetting that (x + q) means the center coordinate is – q.
- Leaving the general equation with a nonzero right side instead of moving all terms to the left.
- Using unequal coefficients for x² and y²; without normalization, that generally describes a different conic rather than a circle.
- Rounding intermediate coefficients too early. Keep full precision and round only the final display.