Equation of a Circle Calculator

The equation of a circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². This calculator writes that equation from the centre and radius, or works backwards from the general form x² + y² + Dx + Ey + F = 0 by completing the square, giving the centre, radius and a graph of the circle.

Try:

How to use the equation of a circle calculator

  1. Choose whether you know the centre and radius, or the general form.
  2. Enter the values.
  3. Press Calculate to see both forms of the equation and the graph.

Formula

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Worked example: Centre (2, −3), r = 5

  • I know: Centre and radius
  • Centre x (h): 2
  • Centre y (k): -3
  • Radius r: 5

✓ Answer checked

(x−2)2+(y+3)2=25(x - 2)^2 + (y + 3)^2 = 25
Centre
(2,  −3)(2,\; -3)
Radius
55
General form
x2+y2−4x+6y−12=0x^2 + y^2 - 4x + 6y - 12 = 0
−8−6−4−224681012−10−8−6−4−224centre (2, -3)
Circle with centre (2, -3) and radius 5.
Step-by-step working (3 steps)

  1. Standard form Equation of a circle

    A circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r².

    (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

  2. Read off centre and radius

    Compare with (x − h)² + (y − k)² = r².

    centre (2,−3),  r=5\text{centre } (2, -3),\; r = 5

  3. General form

    Expand the brackets and move everything to one side.

    x2+y2−4x+6y−12=0x^2 + y^2 - 4x + 6y - 12 = 0
Formulas used
(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
How this was checked
  • ✓ Converting the general form back gives the same centre and radius.

Frequently asked questions

What is the equation of a circle with centre (2, −3) and radius 5?

(x − 2)² + (y + 3)² = 25.

How do I find the centre from the general form?

The centre is (−D/2, −E/2). For x² + y² − 4x + 6y − 12 = 0, that is (2, −3).

How do I find the radius?

r² = h² + k² − F. Then take the square root.

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