BYTETOOLS

Equation of a Circle Calculator

Convert a circle between general form and standard form, or build its equation from a center and radius, a point on the circle, or any three points.

(2, -3)
Center
5
Radius
31.4159
Circumference
78.5398
Area

Both forms of the equation

Standard (center-radius) form
(x - 2)² + (y + 3)² = 25
General form
x² + y² - 4x + 6y - 12 = 0
  • Diameter: 10
  • x-intercepts: -2 and 6
  • y-intercepts: -7.582576 and 1.582576

Sketch

How the conversion works

Completing the square on x² + Dx and y² + Ey moves the center to (-D/2, -E/2) and leaves a radius of √(D²/4 + E²/4 - F). Reversing it, a circle centered at (h, k) expands to x² + y² - 2hx - 2ky + (h² + k² - r²) = 0.

What is the Equation of a Circle Calculator?

The ByteTools Equation of a Circle Calculator moves freely between the two forms every algebra course uses.

  • General form to standard form by completing the square, with the reverse conversion too
  • Builds the equation from a center and radius, a center and a point, or three points
  • Three-point mode solves the 3x3 system and warns when the points are collinear
  • Reports diameter, circumference, area and both axis intercepts
  • Scaled sketch of the circle against the x and y axes
  • Runs entirely in your browser — nothing is uploaded

How to use the Equation of a Circle Calculator

  1. 1

    Pick what you know from the dropdown: general form, center and radius, center and a point, or three points.

  2. 2

    Type the coefficients or coordinates into the numbered fields.

  3. 3

    Read the center, radius, circumference and area in the summary tiles.

  4. 4

    Check the standard and general forms of the equation, plus the intercepts, in the results panel.

  5. 5

    Click Copy results to take the whole solution with you.

About the Equation of a Circle Calculator

The ByteTools Equation of a Circle Calculator moves freely between the two forms every algebra course uses. Give it the general form x² + y² + Dx + Ey + F = 0 and it completes the square to find the center (-D/2, -E/2) and the radius √(D²/4 + E²/4 - F). Give it a center and radius and it expands back to general form.

Two more modes cover the questions that usually follow: build the circle from its center plus any point that lies on it, or from three points, which the tool solves as a 3x3 linear system. Every answer comes with the center, radius, diameter, circumference, area, both x- and y-intercepts, and a scaled sketch on the coordinate plane.

All the algebra runs in your browser with JavaScript. Nothing you type is uploaded or stored, so the calculator is private, instant and keeps working offline once the page has loaded.

Frequently asked questions

How do you convert the general form of a circle to standard form?

Complete the square on the x terms and the y terms separately. For x² + y² + Dx + Ey + F = 0 the center lands at (-D/2, -E/2) and the radius is √(D²/4 + E²/4 - F). For example x² + y² - 4x + 6y - 12 = 0 becomes (x - 2)² + (y + 3)² = 25, a circle of radius 5.

What is the standard equation of a circle?

It is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form is preferred because you can read the center and radius straight off it, with no algebra needed. The calculator prints it for every input mode.

Can three points always define a circle?

Only if they are not collinear and none of them repeat. Three points on a straight line have no finite circle through them, so the 3x3 system has no unique solution. The calculator detects this and tells you instead of returning a nonsense answer.

Why does my equation have no circle?

If D²/4 + E²/4 - F comes out negative there is no real radius, so no real points satisfy the equation. If it is exactly zero the equation describes a single point rather than a circle. Both cases are flagged with an explanation instead of a result.

How do you find where a circle crosses the axes?

Set y = 0 to get x² + Dx + F = 0 and solve the quadratic for the x-intercepts, then set x = 0 and do the same for the y-intercepts. A negative discriminant means the circle never reaches that axis, which the tool states plainly.

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