The same circle can be written in several ways. The standard form shows the centre and radius at a glance; the general form is what you get after expanding; the parametric form is used to draw and animate circles.
How to use the Equation of a Circle Calculator
- Form of the equation
- h (centre x)
- k (centre y)
- Radius r
- D
- E
- F
Results update as you type, and everything is calculated in your browser.
How it works
Choose the form you have. From the general form the calculator completes the square; if h² + k² − F is negative, the equation describes no real circle.
Inputs explained
| Field | What to enter |
|---|---|
| Form of the equation | Choose one: Standard: (x − h)² + (y − k)² = r², General: x² + y² + Dx + Ey + F = 0, Parametric: x = h + r cos t, y = k + r sin t |
| h (centre x) | A number |
| k (centre y) | A number |
| Radius r | A number |
| D | A number |
| E | A number |
| F | A number |
What each result means
- Diameter = 2×[Radius]
- Area = pi×[Radius]^2
- Circumference = 2×pi×[Radius]
- General form D = -2×[Centre x]
- General form E = -2×[Centre y]
- General form F = [Centre x]^2 + [Centre y]^2 – [Radius]^2
Example values
How the main result changes when h changes and everything else stays at the example values above:
| h (centre x) | Radius |
|---|---|
| 1 | 4 |
| 1.5 | 4 |
| 2 | 4 |
| 2.5 | 4 |
| 3 | 4 |
| 4 | 4 |
Formula
Standard: (x − h)² + (y − k)² = r²
General: x² + y² + Dx + Ey + F = 0 with h = −D/2, k = −E/2, r = √(h² + k² − F)
Parametric: x = h + r cos t, y = k + r sin t
Worked example
Inputs — Form of the equation: Standard: (x − h)² + (y − k)² = r²; h (centre x): 2; k (centre y): -3; Radius r: 4.
Results — Centre x = 2; Centre y = -3; Radius = 4; Diameter = 8; Area = 50.265482; Circumference = 25.132741; General form D = -4; General form E = 6; General form F = -3.
Key terms
- Standard form
- (x − h)² + (y − k)² = r².
Frequently asked questions
How do I find the centre from the general form?
h = −D/2, k = −E/2.
What if r² comes out negative?
There is no real circle.
What is the equation of a circle centred at the origin?
x² + y² = r².
How do I get from standard to general form?
Expand the squares and move everything to one side.