Quick answer
A circle with centre (h, k) and radius r has the standard equation (x − h)² + (y − k)² = r², and expanding gives x² + y² + Dx + Ey + F = 0. Centre (1, −2) and radius 3 give (x − 1)² + (y + 2)² = 9, or x² + y² − 2x + 4y − 4 = 0.
Enter the centre coordinates and radius. The calculator returns the standard and general forms.
Related tools: Circle Calculator: Radius, diameter, circumference and area of a circle from any one of them. Area of a Circle Calculator: Area of a circle from its radius, diameter or circumference. Circle Diameter Calculator: Diameter of a circle from radius, circumference or area.
How to use the Circle Equation Calculator
- Enter centre x (h).
- Enter centre y (k).
- Enter radius (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Centre x (h) | A number | — |
| Centre y (k) | A number | — |
| Radius | A number | > 0 |
Formula and method
Formula
(x − h)² + (y − k)² = r²; general form x² + y² + Dx + Ey + F = 0
IEEE-754 double precision; shown to at most 6 decimal places.
Expands the standard form: D = −2h, E = −2k, F = h² + k² − r².
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Centre x (h) = 1; Centre y (k) = -2; Radius = 3.
| Result | Value |
|---|---|
| Standard form | (x − 1)² + (y + 2)² = 9 |
| General form | x² + y² - 2x + 4y - 4 = 0 |
Example 2
Inputs: Centre x (h) = 0; Centre y (k) = 0; Radius = 5.
| Result | Value |
|---|---|
| Standard form | x² + y² = 25 |
| General form | x² + y² - 25 = 0 |
The first line is the headline result (standard form). Below it the circle Equation Calculator lists general form, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Expanding the example
(x − 1)² = x² − 2x + 1 and (y + 2)² = y² + 4y + 4. Adding gives x² + y² − 2x + 4y + 5 = 9, so x² + y² − 2x + 4y − 4 = 0. In general D = −2h, E = −2k and F = h² + k² − r².
Going back
To recover the centre from the general form, complete the square: h = −D/2, k = −E/2 and r² = h² + k² − F. A positive radius is required; r² must be positive for a real circle.
How the result changes with centre x
Holding every other input at the first worked example, this table changes only centre x. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Centre x | Standard form | General form |
|---|---|---|
| 0.5 | (x − 0.5)² + (y + 2)² = 9 | x² + y² - 1x + 4y - 4.75 = 0 |
| 1 | (x − 1)² + (y + 2)² = 9 | x² + y² - 2x + 4y - 4 = 0 |
| 2 | (x − 2)² + (y + 2)² = 9 | x² + y² - 4x + 4y - 1 = 0 |
| 4 | (x − 4)² + (y + 2)² = 9 | x² + y² - 8x + 4y + 11 = 0 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Radius: entering "0" shows "Radius must be greater than 0." beside that field and no result.
Choosing between the Circle Equation Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Circle Calculator](/math/circle-calculator/) | i know the and value | Area |
| [Area of a Circle Calculator](/math/area-of-a-circle/) | i know the and value | Area |
| [Circle Diameter Calculator](/math/circle-diameter/) | i know the and value | Diameter |
| [Circle Perimeter Calculator](/math/circle-perimeter/) | radius | Circle Perimeter |
| [Circle Segment Area Calculator](/math/segment-area/) | radius and central angle | Segment area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Positive radius
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for radius to confirm this.
Frequently asked questions
What is the equation of a circle at the origin?
A circle centred on the origin is x² + y² = r²; r = 3 gives x² + y² = 9.
How do I find the centre from the general form?
h = −D/2 and k = −E/2.
Can the radius be zero?
That is a single point, not a circle.
What does the Circle Equation Calculator calculate?
Equation of a circle from its centre and radius. It returns Standard form and General form.
What formula does the Circle Equation Calculator use?
(x − h)² + (y − k)² = r²; general form x² + y² + Dx + Ey + F = 0. Expands the standard form: D = −2h, E = −2k, F = h² + k² − r².
What do I need to enter in the circle Equation Calculator?
You enter Centre x (h), Centre y (k) and Radius.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Positive radius.
What is an example of a circle Equation result?
With centre x (h) 1, centre y (k) -2 and radius 3, the standard form is (x − 1)² + (y + 2)² = 9.
Related calculators
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
- Area of a Circle Calculator — Area of a circle from its radius, diameter or circumference.
- Circle Diameter Calculator — Diameter of a circle from radius, circumference or area.
- Circle Perimeter Calculator — Perimeter (circumference) of a circle from its radius.
- Circle Segment Area Calculator — Area of a circular segment from radius and central angle.
- Area Calculator — Area of common two-dimensional shapes.
- Regular Polygon Calculator — Area, perimeter and angles of a regular polygon.
- Polygon Interior Angle Calculator — Interior and exterior angles of a polygon.
- Golden Ratio Calculator — Split a length into golden-ratio parts.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.