Goal
Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients.
Circle Equation
Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients.
Key facts
- What it does
- Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients.
- Formula
(x-h)^2+(y-k)^2=r^2, or x^2+y^2+D*x+E*y+F=0 with D=-2h, E=-2k, F=h^2+k^2-r^2.- You enter
- Center h · Center k · Radius r
- Worked example
- (x-2)^2+(y+3)^2=25; expanded form is x^2+y^2-4x+6y-12=0, so D=-4, E=6, and F=-12.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Center h · Center k · Radius r
Method
(x-h)^2+(y-k)^2=r^2, or x^2+y^2+D*x+E*y+F=0 with D=-2h, E=-2k, F=h^2+k^2-r^2.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Circle Equation
Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients.
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Result
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Calculation map
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01
Inputs (3)
- Center h Ready
- Center k Ready
- Radius r Ready
02
Formula
(x-h)^2+(y-k)^2=r^2, or x^2+y^2+D*x+E*y+F=0 with D=-2h, E=-2k, F=h^2+k^2-r^2.
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
Recent runs
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Formula, assumptions, and example
Formula: (x-h)^2+(y-k)^2=r^2, or x^2+y^2+D*x+E*y+F=0 with D=-2h, E=-2k, F=h^2+k^2-r^2.
The center shifts the two coordinate axes and the positive radius sets the squared distance from the center. Expansion supplies the coefficients of x and y and the constant term in the implicit quadratic equation.
- The circle lies in a flat two-dimensional Cartesian plane with one shared coordinate unit.
- The center coordinates are finite and the radius is strictly positive; no degenerate point-circle is accepted.
- The returned coefficients describe the equation only and do not calculate area, circumference, intersections, or a fitted circle.
Worked example: (x-2)^2+(y+3)^2=25; expanded form is x^2+y^2-4x+6y-12=0, so D=-4, E=6, and F=-12.
Displayed input contract
- Center h · minimum -1000000 · maximum 1000000
- Center k · minimum -1000000 · maximum 1000000
- Radius r · minimum 1.0E-6 · maximum 1000000
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Circle Equation for a real question
Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes circle equation, standard circle form, center radius. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Center h · Center k · Radius r. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. The circle lies in a flat two-dimensional Cartesian plane with one shared coordinate unit.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Circle Equation
- Enter Center h — The x-coordinate of the circle center. (coordinate units).
- Enter Center k — The y-coordinate of the circle center. (coordinate units).
- Enter Radius r — A strictly positive circle radius. (length units).
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
(x-h)^2+(y-k)^2=r^2, or x^2+y^2+D*x+E*y+F=0 with D=-2h, E=-2k, F=h^2+k^2-r^2.
The center shifts the two coordinate axes and the positive radius sets the squared distance from the center. Expansion supplies the coefficients of x and y and the constant term in the implicit quadratic equation.
Worked example
(x-2)^2+(y+3)^2=25; expanded form is x^2+y^2-4x+6y-12=0, so D=-4, E=6, and F=-12.
Assumptions and limits
- The circle lies in a flat two-dimensional Cartesian plane with one shared coordinate unit.
- The center coordinates are finite and the radius is strictly positive; no degenerate point-circle is accepted.
- The returned coefficients describe the equation only and do not calculate area, circumference, intersections, or a fitted circle.
Who uses this calculator?
- Students practicing center-radius and general quadratic forms
- Teachers demonstrating completing-the-square relationships
- Developers storing circles as implicit equations
When is it useful?
- Convert a center-radius description into a standard equation.
- Obtain D, E, and F for a later implicit-circle calculation.
- Check signs when expanding a translated circle.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
A circle equation describes the set of plane points at one fixed positive distance from a center. This calculator starts with center coordinates h and k and radius r, returns the translated standard form, and also supplies the expanded coefficients in x^2+y^2+D*x+E*y+F=0. The equation-building task is intentionally separate from calculating circle area or circumference: the output is an implicit relation, not a measure of the disk or its boundary. The guide explains translation signs, expansion, worked substitutions, finite validation, degenerate boundaries, related tools, source metadata, assumptions, and conservative limits so a familiar formula is not overextended into a broader geometry claim.
The equation rather than a circle measure
The main output is a relation that can be tested by inserting any point (x, y). Points on the circle satisfy the relation exactly in the mathematical model; points inside or outside produce different squared distances from the center. The calculator returns the standard equation first because center-radius form keeps the geometry visible. It then returns the radius squared and the three expanded coefficients D, E, and F so another bounded calculation can use the implicit representation without repeating the algebra.
This contract is distinct from a circle area or circumference calculator. Area concerns the filled disk and circumference concerns the boundary length. An equation instead describes membership in a locus. A positive radius is required even though a radius of zero could describe one point in a limiting sense. Rejecting that degenerate case preserves the ordinary circle interpretation and makes the output assumptions match the title.
- The result is an implicit locus equation.
- Standard form keeps center and radius visible.
- Expanded coefficients support a separate quadratic representation.
- Area and circumference are intentionally not calculated here.
Enter center and radius
The h field is the center's x-coordinate and k is its y-coordinate. A positive h moves the center right and a negative h moves it left; the same sign interpretation applies to k vertically. Radius r is a length and must be greater than zero. All three fields are finite numbers within the displayed software bounds. The form has no unit conversion selector, so coordinates and radius should be expressed in one compatible length convention before entry.
Direct calls use the same strict type and range checks as form submissions. Numeric strings, blanks, NaN, infinity, and values outside the declared bounds are rejected. The radius lower bound is a small positive software boundary rather than a universal geometric minimum. Center coordinates may be zero or negative. A zero center is ordinary; it means the circle is centered at the origin and simplifies the translated terms without changing the formula.
- h and k locate the center in the Cartesian plane.
- r is a strictly positive length.
- Negative center coordinates are valid.
- No automatic unit conversion or expression parsing occurs.
Standard form from distance geometry
The Euclidean distance from a point (x, y) to the center (h, k) is sqrt((x-h)^2+(y-k)^2). Setting that distance equal to r and squaring both sides gives (x-h)^2+(y-k)^2=r^2. Squaring is safe here because both sides of the original distance equation are nonnegative. This form exposes the horizontal and vertical translation directly. The calculator computes r^2 as a finite numeric result and inserts it into the equation text.
The parentheses matter. If k is negative, y-k becomes y+|k|, not y-|k|. The helper that builds the equation chooses the visible sign from the entered center while the numeric coefficient calculation uses the original signed value. Keeping those two layers aligned prevents a readable equation from disagreeing with D and E. The equation text is a renderer-safe string; it is not subsequently parsed or evaluated by the engine.
- Distance from the center gives the standard relation.
- Both translated coordinate terms are squared.
- A negative center coordinate changes the visible sign inside parentheses.
- The equation string is descriptive output, not executable input.
Expansion and coefficient meanings
Expanding gives x^2-2h*x+h^2+y^2-2k*y+k^2=r^2. Move the right side to the left to obtain x^2+y^2-2h*x-2k*y+(h^2+k^2-r^2)=0. The requested coefficients are D=-2h, E=-2k, and F=h^2+k^2-r^2. The quadratic coefficients of x^2 and y^2 are both one, and there is no xy term. That pattern is the algebraic signature of an axis-aligned circle in ordinary Cartesian coordinates.
The signs of D and E are tied directly to the center. A center at h=2 produces D=-4, while k=-3 produces E=6. F combines the squared center distance with the negative radius squared, so it can be positive, zero, or negative. These coefficients are not independent free inputs in this calculator; they are derived from the center-radius fields. That prevents an inconsistent general quadratic from being mislabeled as a circle.
- D=-2h and E=-2k.
- F=h^2+k^2-r^2.
- The x^2 and y^2 coefficients are both one.
- There is no xy term in this axis-aligned circle form.
A worked center-radius example
Use h=2, k=-3, and r=5. The translated form is (x-2)^2+(y+3)^2=25. The radius squared is 25. Expanding gives D=-4, E=6, and F=2^2+(-3)^2-25=4+9-25=-12, so the general form is x^2+y^2-4x+6y-12=0. The center and radius remain visible in the first equation, while the coefficient results make the same locus available to an implicit-equation consumer.
Check the center point (2,-3): the left side of the standard form is zero, so the center is inside rather than on the circle. Check (2,2), which is five units vertically from the center: (2-2)^2+(2+3)^2=25, so it lies on the boundary. Check (2,3), six units away: its squared distance is 36 and it lies outside. These substitutions clarify that the equation describes a boundary, not a filled region.
- The example produces (x-2)^2+(y+3)^2=25.
- The expanded constant is F=-12.
- A point five units from the center is on the circle.
- A point farther than five units is outside the boundary.
Center signs and translated parentheses
The standard form always uses x-h and y-k. When h is positive, the visible x term is x minus that positive value. When h is negative, the visible term becomes x plus its magnitude. The same rule applies independently to k. For h=-4 and k=6, the equation contains (x+4)^2+(y-6)^2. Reading the sign inside the parentheses as the opposite of the center coordinate is a reliable way to avoid an expansion error.
The numeric coefficients use the signed center directly: D=-2h and E=-2k. Thus h=-4 gives D=8, and k=6 gives E=-12. The helper that formats the equation and the arithmetic that derives coefficients must agree on that sign relationship. A visually plausible equation with a mismatched D or E would describe a different center, which is why the focused tests should inspect both text and numeric coefficient results.
- The sign inside each parenthesis is opposite its center coordinate.
- D and E retain the signed center relationship.
- Horizontal and vertical translations are independent.
- Check equation text against coefficient signs.
What the radius controls
Radius r is the fixed distance from the center to every point on the circle. The equation uses r^2, so changing r changes the right side and F while leaving D and E unchanged. A larger radius produces a larger boundary around the same center. A smaller positive radius produces a smaller boundary. The calculator returns r^2 as a numeric result because it is central to the standard equation, but it does not infer area, circumference, diameter, or a unit conversion.
The radius field is bounded below by a small positive value and above by the shared positive limit. A radius of zero would collapse the locus to one point. Although such a set can be studied algebraically, this record is intentionally about an ordinary circle with a positive size. Negative radius values do not add a new circle because distance is nonnegative; they are rejected rather than converted to their absolute value so the input contract stays explicit.
- r sets the fixed center-to-boundary distance.
- r^2 appears on the right side of standard form.
- Changing r does not change D or E.
- Zero and negative radii are outside this ordinary-circle contract.
Testing points against the equation
For a candidate point (u, v), evaluate (u-h)^2+(v-k)^2. Equality with r^2 means the point is on the circle. A value less than r^2 means the point is inside the disk, and a value greater than r^2 means it is outside. This classification is an interpretation step; the handler itself constructs the equation and does not accept a third point for membership testing. Keeping the tasks separate prevents a circle equation page from silently becoming a point-in-region tool.
The expanded form gives the same classification after evaluating u^2+v^2+D*u+E*v+F, which should equal the standard expression minus r^2. Because decimal arithmetic is finite precision, a caller comparing a result to zero should use a tolerance appropriate to its scale. The calculator does not promise exact symbolic equality for arbitrary decimal substitutions. Its finite checks guarantee safe numeric construction, not unlimited precision.
- Equality identifies the boundary.
- Less than r^2 is inside the disk.
- Greater than r^2 is outside the disk.
- Decimal membership checks need an appropriate tolerance.
Circle equation versus nearby models
A circle equation is not the same as a circle area calculator. Area is pi*r^2 and describes the filled disk, while the equation describes the points at exactly radius r. It is also distinct from circumference, which measures boundary length. The catalog keeps these questions separate so the title, fields, formula, and output all agree. A visitor who needs more than the equation should choose a separate calculator rather than reinterpret the coefficient fields.
This page also excludes rotated conics and general quadratic fitting. A quadratic with unequal x^2 and y^2 coefficients, an xy term, or noisy point data may describe an ellipse, a rotated conic, or a fitted model. The present inputs guarantee a center-radius circle in an axis-aligned Cartesian plane. The formula is reusable, but the narrow contract is a feature: it prevents a familiar equation from being treated as a universal curve classifier.
- Equation, area, and circumference answer different questions.
- The model is axis-aligned in Cartesian coordinates.
- Rotated conics and general quadratic fitting are excluded.
- Do not infer a region or measurement from equation coefficients alone.
Finite validation and renderer safety
The pure handler requires h, k, and r to be JavaScript numbers that are finite and within the displayed bounds. It checks r^2, D, E, and F before constructing result entries. A direct call with a numeric string, blank, NaN, infinity, or an out-of-range value throws instead of relying on browser form coercion. The same validation makes the module usable in focused tests and in the browser registry without a hidden second implementation of the domain rules.
The equation is returned as text and the derived measures are returned as finite numeric results with labels and units. No expression is parsed after it is formatted, so symbols such as h and k remain explanatory text. Negative zero is normalized in numeric output. The renderer can show the standard equation, radius squared, and coefficients without deciding whether the user intended an area, a circumference, or an intersection operation.
- All three inputs are finite-checked.
- The radius is bounded strictly above zero.
- Derived coefficients are checked before rendering.
- Text equations are descriptive and not reparsed.
Units, coordinate frames, and limits
If h, k, and r are in metres, the center coordinates and radius are in metres and the squared radius and F have square-length implications in the equation. If the values are abstract graph units, the outputs retain that abstraction. The calculator has no unit selector and cannot know whether a coordinate was measured, converted, or copied from a drawing. Use one origin, axis orientation, scale, and unit convention for the center and radius before calculation.
The plus or minus 1,000,000 coordinate bound and positive radius bound are conservative software limits. They keep intermediate squares and formatted outputs finite for ordinary browser use, but they do not define the largest possible circle in mathematics. Very large or highly precise physical inputs may need a numerical library, uncertainty analysis, or unit scaling outside this handler. Display precision is not a measurement guarantee and should not be used as one.
- Center and radius must use compatible length units.
- No unit conversion is performed.
- Bounds are software safeguards, not mathematical limits.
- Precision and measurement uncertainty remain external concerns.
Source boundary and conservative interpretation
The catalog source is private formula provenance for the standard center-radius and expanded circle forms. This public article is original WorldCalculate writing and does not reproduce another site's body, code, branding, defaults, or calculator data. Formula provenance confirms the algebraic relationship, but it cannot confirm that a visitor's center and radius came from a valid survey, drawing, sensor, or design specification. That provenance and review belong to the surrounding workflow.
The model assumes a flat Euclidean plane, finite exact-enough inputs, a positive radius, and an axis-aligned standard form. It excludes uncertainty bands, fitted circles, perspective correction, geographic projection, rotated conics, and physical tolerance decisions. A circle equation can be a useful idealization while a real object has thickness or deformation. Use the result as a transparent equation construction, then apply domain-specific validation before relying on it.
- Source metadata supports the formula, not the visitor's measurements.
- The plane and coordinate frame must be declared externally.
- Fitting and perspective correction are separate models.
- An ideal circle equation is not a tolerance or manufacturing decision.
Frequently asked questions
Why does a negative center make a plus sign? The form is x-h, so a negative h becomes x+|h|; the expanded coefficient still follows D=-2h. Why is radius required to be positive? This page models an ordinary nondegenerate circle rather than a single point. Does F identify the radius? Not by itself: F also contains h^2+k^2, so center and radius fields are needed to interpret it. Why is there no xy term? The contract is an unrotated circle in standard Cartesian axes.
Can I use the equation to compute area? You can use the same radius in a separate area formula, but this handler intentionally returns equation outputs only. Can I enter a general quadratic? No. The fields describe center and positive radius, not free polynomial coefficients. What if an observed shape is almost circular? The calculator can provide an ideal reference circle, but fitting and residual analysis should be done by a method that accepts observations and uncertainty.
- Parenthesis signs reverse the signed center values.
- F is derived and is not a standalone radius field.
- A general quadratic is outside the input contract.
- Area, fitting, and uncertainty need separate calculations.
Circle transformations and invariants
Changing the center translates the circle without changing its radius or shape. If h and k are both increased by known offsets, every point on the circle moves by those offsets, while r^2 remains unchanged. Changing the radius scales the boundary around the same center. These simple transformations help review a result: D and E should respond to center changes, F should respond to center and radius changes, and the standard equation should retain the same parenthesis structure.
A rotation of a circle about its center does not change the geometric circle, which is why no orientation field is needed. A coordinate transformation with unequal scale, however, can turn a circle into an ellipse in the new coordinates. The calculator assumes the entered frame is already the intended Euclidean frame. It does not infer whether a drawing was stretched, projected, or converted from another system.
- Translation changes center terms but not radius.
- Radius changes alter r^2 and F.
- A circle has no orientation parameter in this model.
- Unequal coordinate scaling can require an ellipse model.
Worked coefficient and membership review
For h=2, k=-3, and r=5, verify the expanded expression at the boundary point (2,2). The standard left side is (2-2)^2+(2+3)^2=25. In the expanded form, 2^2+2^2-4(2)+6(2)-12 equals 4+4-8+12-12=0. The two forms therefore agree at that point. Repeating the substitution at (7,-3) gives the same boundary result and tests the horizontal translation separately.
For a point such as (2,3), the standard left side is 36, which is greater than 25 and confirms it is outside. For the center (2,-3), the left side is zero and confirms it is inside. These tests are not returned by the handler because no test-point fields exist. They are useful review procedures and explain why an equation result should not be confused with a single boolean membership answer.
- Boundary substitution should make the expanded form zero.
- The standard and expanded equations should agree.
- The center is inside the circle.
- A test point is external to this three-field handler.
Preserving equation meaning in a handoff
A downstream graph, solver, or document should retain whether the equation is the standard center-radius form or its expanded coefficient form. The two strings describe the same ideal boundary when D, E, and F were derived from h, k, and r, but the standard form makes the geometry easier to audit. Include the positive-radius assumption, the coordinate unit, and the fact that the equation represents a boundary rather than the filled disk.
If a later calculation needs a point test, area, circumference, intersection, or fitted curve, add that operation explicitly. Do not infer it from the presence of D, E, and F. The pure handler is intentionally small and deterministic: it constructs one circle relation, returns finite derived coefficients, and leaves application-specific questions to their own contracts. This separation helps prevent a correct equation from being used as an unexplained physical or statistical conclusion.
- Keep standard and expanded forms linked.
- Retain radius, center, units, and boundary meaning.
- Point tests and measures are separate operations.
- Do not infer application conclusions from coefficients alone.
Frequently asked questions
What is the Circle Equation?
Build the standard circle equation from a center and positive radius, then return the expanded quadratic coefficients.
What is the formula for the Circle Equation?
(x-h)^2+(y-k)^2=r^2, or x^2+y^2+D*x+E*y+F=0 with D=-2h, E=-2k, F=h^2+k^2-r^2. The center shifts the two coordinate axes and the positive radius sets the squared distance from the center. Expansion supplies the coefficients of x and y and the constant term in the implicit quadratic equation.
What do I need to use this calculator?
Enter Center h, Center k, Radius r, then choose Calculate.
What are the limits of this calculator?
The circle lies in a flat two-dimensional Cartesian plane with one shared coordinate unit. The center coordinates are finite and the radius is strictly positive; no degenerate point-circle is accepted. The returned coefficients describe the equation only and do not calculate area, circumference, intersections, or a fitted circle.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.