Calculator
White themeThis tool converts a circle in general form a·x² + a·y² + d·x + e·y + f = 0 to standard form (x − h)² + (y − k)² = r². For a valid circle, a ≠ 0.
Results
Normalized General Form
x² + y² + Dx + Ey + F = 0
D = —,
E = —,
F = —
Standard Form
(x − h)² + (y − k)² = r²
h = —, k = —, r² = —, r = —
Completed Square Derivation
—
Example Data & Exports
| a | d | e | f | D | E | F | h | k | r² | r | Standard Form |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | -6 | 4 | -3 | -6 | 4 | -3 | 3 | -2 | 16 | 4 | (x - 3)² + (y + 2)² = 16 |
| 2 | 8 | -10 | -26 | 4 | -5 | -13 | -2 | 2.5 | 23.25 | 4.821825 | (x + 2)² + (y - 2.5)² = 23.25 |
| 1 | 0 | 0 | -9 | 0 | 0 | -9 | 0 | 0 | 9 | 3 | (x - 0)² + (y - 0)² = 9 |
How to use this calculator
- Enter a, d, e, f for a·x² + a·y² + d·x + e·y + f = 0.
- Click Calculate to get D, E, F, center, and radius.
- Use Calculate & Save Row to append a result to the table.
- Export the accumulated table with Download CSV or capture a report with Download PDF.
- If a = 0 or r² ≤ 0, the input is not a proper circle.
Formulas used
Normalize by dividing through by a: x² + y² + Dx + Ey + F = 0, where D = d/a, E = e/a, F = f/a.
Then center and radius are: h = −D/2, k = −E/2, r² = h² + k² − F, r = √(r²).
Derivation via completing the square:
x² + Dx + y² + Ey + F = 0
(x² + Dx) + (y² + Ey) = −F
(x + D/2)² − (D/2)² + (y + E/2)² − (E/2)² = −F
(x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F
(x − h)² + (y − k)² = r², with h = −D/2, k = −E/2.
(x² + Dx) + (y² + Ey) = −F
(x + D/2)² − (D/2)² + (y + E/2)² − (E/2)² = −F
(x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F
(x − h)² + (y − k)² = r², with h = −D/2, k = −E/2.
Notes & validation
- Requires a ≠ 0. If r² ≤ 0, no real circle exists.
- This calculator assumes no xy term and equal quadratic coefficients.
- Rounding uses 6 significant decimals for readability in displays.