acalculator

What is the equation of a circle?

Type the center and radius, the center and a point on the circle, the two ends of a diameter, or the coefficients of the general form. The equation of a circle calculator writes both forms and finds the center and radius.

Your numbers

Standard form
(x + 1)² + (y − 3)² = 4

The equation of the circle is (x + 1)² + (y − 3)² = 4.

General form
x² + y² + 2x − 6y + 6 = 0
Center (h, k)
(−1, 3)
Radius r
2
r²
4
Diameter
4
Circumference
12.56637061
Area
12.56637061
Working
r² = 2² = 4; standard form (x + 1)² + (y − 3)² = 4; expand: D = −2h = 2, E = −2k = −6, F = h² + k² − r² = 6; r = √4 = 2

Standard form: (x + 1)² + (y − 3)² = 4. The equation of the circle is (x + 1)² + (y − 3)² = 4.

How it is worked out

How to calculate

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.

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.

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.

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. I know Center and radius, Center x (h) -1, Center y (k) 3, Radius (r) 2 gives Standard form (x + 1)² + (y − 3)² = 4, General form x² + y² + 2x − 6y + 6 = 0, Radius r 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. I know Center and radius, Center x (h) 0, Center y (k) 0, Radius (r) 3 gives Standard form x² + y² = 9, General form 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. I know Center and a point, Center x (h) 2, Center y (k) 4, Point x -2, Point y 1 gives Standard form (x − 2)² + (y − 4)² = 25, Radius r 5, General form 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. I know General form, D (x coefficient) -4, E (y coefficient) -6, F (constant) 4 gives Standard form (x − 2)² + (y − 3)² = 9, Center (h, k) (2, 3), Radius r 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. I know Ends of a diameter, First end x 0, First end y 0, Second end x 2, Second end y 2 gives Standard form (x − 1)² + (y − 1)² = 2, Radius r 1.414214, General form x² + y² − 2x − 2y = 0.

How it works

Standard form. A circle with center (h, k) and radius r is every point (x, y) at distance r from the center: (x − h)² + (y − k)² = r².

General form. Expanding gives x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k and F = h² + k² − r².

The page first finds the center (h, k) and r², then writes both forms:

  • Center and radius: r² = r × r.
  • Center and a point (px, py): r² = (px − h)² + (py − k)².
  • Ends of a diameter (x₁, y₁) and (x₂, y₂): h = (x₁ + x₂) ÷ 2, k = (y₁ + y₂) ÷ 2, r² = (x₁ − h)² + (y₁ − k)².
  • General form (completing the square): h = −D ÷ 2, k = −E ÷ 2, r² = h² + k² − F.

From the radius: diameter 2r, circumference 2πr, area πr².

Rules

  • Every number is from −10¹² to 10¹²; the radius is more than 0.
  • A point equal to the center, two equal diameter ends, or a general form with r² ≤ 0 gives no answer.
  • How the forms are written: (x − h)² becomes (x + |h|)² when h is negative and x² when h is 0; in the general form a term with coefficient 0 is left out, a coefficient of 1 or −1 on x or y is written as the bare letter, and a negative term is written with "−".

Output format. Typed decimals are read exactly, so h, k, r², D, E and F are exact, each rounded once to 10 significant figures. The radius is exact when r² is the square of a fraction; otherwise it is the double-precision square root of r². Diameter, circumference and area use that radius and double-precision π.

Worked examples by hand

Center (−1, 3), radius 2. r² = 4. Standard form (x + 1)² + (y − 3)² = 4. D = 2, E = −6, F = 1 + 9 − 4 = 6: x² + y² + 2x − 6y + 6 = 0.

Center (0, 0), radius 3. x² + y² = 9, general form x² + y² − 9 = 0. Area 9π = 28.2743.

Center (2, 4), point (−2, 1). r² = (−4)² + (−3)² = 25, r = 5. (x − 2)² + (y − 4)² = 25. D = −4, E = −8, F = 4 + 16 − 25 = −5: x² + y² − 4x − 8y − 5 = 0.

General form x² + y² − 4x − 6y + 4 = 0. h = 2, k = 3, r² = 4 + 9 − 4 = 9, r = 3. (x − 2)² + (y − 3)² = 9.

Diameter from (0, 0) to (2, 2). Center (1, 1), r² = 1 + 1 = 2, r = √2 = 1.4142. (x − 1)² + (y − 1)² = 2; general form x² + y² − 2x − 2y = 0.

Other questions people ask

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.