# Equation of a Circle Calculator

URL: https://speedycalculator.com/calculators/math/circle-equation/
Last updated: 2026-10-30

> The equation of a circle calculator converts between the standard form (x − h)² + (y − k)² = r², the general form x² + y² + Dx + Ey + F = 0 and the parametric form, and gives the centre, radius, diameter, area and circumference.

## Inputs

- Form of the equation
- h (centre x)
- k (centre y)
- Radius r
- D
- E
- F

## Results

- Centre x
- Centre y
- Radius
- Diameter
- Area
- Circumference
- General form D
- General form E
- General form F

## 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
```

## Explanation

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

FieldWhat to enterForm of the equationChoose 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 th (centre x)A numberk (centre y)A numberRadius rA numberDA numberEA numberFA 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)Radius141.54242.543444

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Circle](https://mathworld.wolfram.com/Circle.html)