# What is the equation of a circle?

Writes the standard and general form of a circle’s equation from its center and radius, a center and a point, the ends of a diameter, or the general form, with the center and radius.

- Page: https://www.acalculator.org/math/equation-of-a-circle-calculator
- JSON spec: https://www.acalculator.org/math/equation-of-a-circle-calculator.json
- Version: 35ccb25731d9

## Default answer

Example with the default inputs (I know Center and radius, Center x (h) −1, Center y (k) 3, Radius (r) 2): The equation of the circle is (x + 1)² + (y − 3)² = 4.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| from | I know | What you know about the circle. |
| h | Center x (h) | The x coordinate of the center. |
| k | Center y (k) | The y coordinate of the center. |
| r | Radius (r) | The distance from the center to the circle. |
| px | Point x | The x coordinate of a point on the circle. |
| py | Point y | The y coordinate of a point on the circle. |
| x1 | First end x | The x coordinate of one end of a diameter. |
| y1 | First end y | The y coordinate of one end of a diameter. |
| x2 | Second end x | The x coordinate of the other end of the diameter. |
| y2 | Second end y | The y coordinate of the other end of the diameter. |
| d | D (x coefficient) | D in x² + y² + Dx + Ey + F = 0. |
| e | E (y coefficient) | E in x² + y² + Dx + Ey + F = 0. |
| f | F (constant) | F in x² + y² + Dx + Ey + F = 0. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| standard | Standard form | The equation (x − h)² + (y − k)² = r². |
| general | General form | The equation x² + y² + Dx + Ey + F = 0. |
| center | Center (h, k) | The center of the circle. |
| radius | Radius r | The radius of the circle. |
| rsq | r² | The radius squared, the right side of the standard form. |
| diameter | Diameter | Twice the radius. |
| circumference | Circumference | The distance around the circle, 2πr. |
| area | Area | The space inside the circle, πr². |
| steps | Working | How the center and r² were found. |

## Method

Standard form (x − h)² + (y − k)² = r². General form x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k, F = h² + k² − r²; back by completing the square, h = −D ÷ 2, k = −E ÷ 2, r² = h² + k² − F.

## Assumptions

- Typed decimals are read exactly, so the center, r², D, E and F are exact and rounded once to 10 significant figures.
- The radius is exact when r² is a perfect square of a fraction; otherwise it is the double-precision √(r²).
- A general form with r² ≤ 0 is a point or has no points, so it gives no answer.

## Worked examples

1. from = center, h = -1, k = 3, r = 2 gives standard = (x + 1)² + (y − 3)² = 4, general = x² + y² + 2x − 6y + 6 = 0, radius = 2. Source: Circles (standard form (x − h)² + (y − k)² = r², general form x² + y² + ax + by + c = 0; Examples 11.5, 11.6, 11.7 and 11.10), https://openstax.org/books/intermediate-algebra-2e/pages/11-1-distance-and-midpoint-formulas-circles (retrieved 2026-10-05) (Example 11.6).
2. from = center, h = 0, k = 0, r = 3 gives standard = x² + y² = 9, general = x² + y² − 9 = 0, area = 28.274334. Source: Circles (standard form (x − h)² + (y − k)² = r², general form x² + y² + ax + by + c = 0; Examples 11.5, 11.6, 11.7 and 11.10), https://openstax.org/books/intermediate-algebra-2e/pages/11-1-distance-and-midpoint-formulas-circles (retrieved 2026-10-05) (Example 11.5).
3. from = point, h = 2, k = 4, px = -2, py = 1 gives standard = (x − 2)² + (y − 4)² = 25, radius = 5, general = x² + y² − 4x − 8y − 5 = 0. Source: Circles (standard form (x − h)² + (y − k)² = r², general form x² + y² + ax + by + c = 0; Examples 11.5, 11.6, 11.7 and 11.10), https://openstax.org/books/intermediate-algebra-2e/pages/11-1-distance-and-midpoint-formulas-circles (retrieved 2026-10-05) (Example 11.7).
4. from = general, d = -4, e = -6, f = 4 gives standard = (x − 2)² + (y − 3)² = 9, center = (2, 3), radius = 3. Source: Circles (standard form (x − h)² + (y − k)² = r², general form x² + y² + ax + by + c = 0; Examples 11.5, 11.6, 11.7 and 11.10), https://openstax.org/books/intermediate-algebra-2e/pages/11-1-distance-and-midpoint-formulas-circles (retrieved 2026-10-05) (Example 11.10).
5. from = diameter, x1 = 0, y1 = 0, x2 = 2, y2 = 2 gives standard = (x − 1)² + (y − 1)² = 2, radius = 1.414214, general = x² + y² − 2x − 2y = 0.

## FAQ

### What is the standard form of the equation of a circle?

(x − h)² + (y − k)² = r², where (h, k) is the center and r the radius. A circle with center (−1, 3) and radius 2 is (x + 1)² + (y − 3)² = 4.

### What is the general form of a circle?

x² + y² + Dx + Ey + F = 0. Expanding the standard form gives D = −2h, E = −2k and F = h² + k² − r². For center (−1, 3) and radius 2 that is x² + y² + 2x − 6y + 6 = 0.

### How do I find the center and radius from the general form?

Complete the square in x and in y. The center is (−D ÷ 2, −E ÷ 2) and r² = h² + k² − F. For x² + y² − 4x − 6y + 4 = 0 the center is (2, 3) and r² = 4 + 9 − 4 = 9, so r = 3.

### How do I write the equation from the center and a point on the circle?

The radius is the distance from the center to the point. For center (2, 4) and point (−2, 1), r² = (−4)² + (−3)² = 25, so the equation is (x − 2)² + (y − 4)² = 25.

### How do I write the equation from the ends of a diameter?

The center is the midpoint of the two ends, and r² is the squared distance from the center to either end. For (0, 0) and (2, 2), the center is (1, 1) and r² = 2, so (x − 1)² + (y − 1)² = 2.

### When is x² + y² + Dx + Ey + F = 0 not a circle?

When h² + k² − F is 0 or less. At 0 the equation holds for one point only (the center); below 0 no point fits. The page then gives no answer.

## Sources

- OpenStax, Intermediate Algebra 2e, §11.1 Distance and Midpoint Formulas; Circles (standard and general form; Examples 11.5, 11.6, 11.7 and 11.10), CC BY 4.0. https://openstax.org/books/intermediate-algebra-2e/pages/11-1-distance-and-midpoint-formulas-circles (retrieved 2026-10-05)