This circle equation calculator converts a center and radius into both forms of the circle’s equation — standard (center-radius) form and expanded general form.

Standard form –
General form –

Standard form (x − h)² + (y − k)² = r² displays the geometry at a glance. Expanding it yields general form x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k, F = h² + k² − r² — the form algebra problems hand back to you to decode by completing the square.

One Circle, Two Costumes: Standard Form Shows, General Form Hides

Area and circumference of the same circle live on our Circle Calculator. Part of our Math Calculators collection.

Worked Example (Verified)

  • Center (2, −3), r = 5 (defaults): standard (x − 2)² + (y + 3)² = 25; general x² + y² − 4x + 6y − 12 = 0 (F = 4 + 9 − 25 = −12)
  • Sign trap, defused: center k = −3 appears in standard form as (y + 3)² — the formula subtracts k, and subtracting a negative adds
  • Decoding the return trip: from −4x, center h = −(−4)/2 = 2; from +6y, k = −3 — halving and negating the linear coefficients recovers the center
  • r² is not r: the standard form’s right side is 25 for a radius-5 circle — the commonest transcription error in the unit

Reverse-Engineering a General Form

  • Complete the square on x and y separately; gather constants on the right.
  • If the resulting right side is negative, no real circle exists (imaginary radius); zero means a single point.
  • Unequal x²/y² coefficients signal an ellipse, not a circle — see the Ellipse Calculator.
  • The equation encodes distance: every (x, y) on the circle is exactly r from (h, k) by Pythagoras.

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Frequently Asked Questions

What is the standard equation of a circle?

(x − h)² + (y − k)² = r², where (h, k) is the center and r the radius. For center (2, −3) and r = 5: (x − 2)² + (y + 3)² = 25. It is the distance formula applied to every point on the rim.

How do I convert standard form to general form?

Expand the squares and collect: x² + y² − 2hx − 2ky + (h² + k² − r²) = 0. The default circle expands to x² + y² − 4x + 6y − 12 = 0. This calculator prints both forms together.

How do I find the center from a general equation?

Complete the square, or shortcut: center = (−D/2, −E/2) from x² + y² + Dx + Ey + F = 0, and r = √(h² + k² − F). Coefficients −4 and +6 decode to center (2, −3), as in the worked example.

Why does (y + 3) mean the center is at −3?

Standard form subtracts the center coordinate: (y − k) with k = −3 evaluates to (y + 3). The sign in the printed equation is always the opposite of the center’s coordinate — the single most-tested trap in this topic.

What if the equation’s right side comes out negative?

Then the “circle” has no real points: the radius squared is negative. Zero gives a degenerate circle — one point, the center. Both cases arise when decoding general forms with completing the square.

Is this circle equation calculator free?

Yes — free, instant, no sign-up.