Circle Equation Calculator

Background.

Every point on a circle sits the same distance from the centre, and the equation of a circle is nothing more than that sentence written with the distance formula. If the centre is (h, k) and the radius is r, a point (x, y) lies on the circle exactly when (x − h)² + (y − k)² = r². Pythagoras supplies the rest: the horizontal and vertical gaps from the centre are the legs of a right triangle whose hypotenuse is the radius.

That centre-radius version, usually called the standard form, is the one you can read at a glance. Algebra courses and graphing software also use the general form, x² + y² + Dx + Ey + F = 0, which is what you get by multiplying out the brackets. It hides the centre and radius but is easier to combine with other equations, for instance when solving for where a line crosses the circle.

Going from standard to general form is mechanical. D is −2h and E is −2k, so they come straight from the centre; F, the constant term, takes a little more work because it collects h², k² and −r² into one number. Students most often slip on signs here: a centre at k = −3 becomes (y + 3)² in the standard form and +6y in the general form.

The constant F has a useful meaning of its own. It is the value the left-hand side takes at the origin, so a negative F means the origin lies inside the circle, zero means the circle passes through it, and a positive F means the origin is outside.

What is circle equation calculator?

The equation of a circle is the condition a point (x, y) must satisfy to lie on it. In standard form, (x − h)² + (y − k)² = r², the centre (h, k) and radius r appear directly, and the right-hand side is the radius squared. Expanding gives the general form x² + y² − 2hx − 2ky + (h² + k² − r²) = 0, whose constant term F = h² + k² − r² is the second number reported here. Only real circles with a positive radius are handled.

How to use this calculator.

1. 01Enter the x-coordinate of the centre as h and the y-coordinate as k, keeping their signs. A centre at (2, −3) means h = 2 and k = −3.
2. 02Enter the radius r as a positive length. If you only know the diameter, halve it; if you know a point on the circle, the radius is its distance from the centre.
3. 03Read r², the right-hand side of the standard form, and write (x − h)² + (y − k)² = r² with your own numbers, flipping each minus to a plus where h or k is negative.
4. 04For the general form, write D = −2h and E = −2k yourself, then take F from the result: x² + y² + Dx + Ey + F = 0.

The formula.

The standard form (x − h)² + (y − k)² = r² is the distance formula with the distance fixed at r: the squared horizontal gap plus the squared vertical gap equals the squared radius. Expanding each bracket gives x² − 2hx + h² and y² − 2ky + k². Moving r² to the left and collecting the constants yields x² + y² − 2hx − 2ky + h² + k² − r² = 0. The two outputs are r² and F = h² + k² − r². Because h² + k² is the squared distance from the origin to the centre, F is positive whenever the centre lies further from the origin than the radius, and a circle centred at the origin always has F = −r². The coefficients of x² and y² are both 1; any equation where they differ describes an ellipse or another conic, not a circle.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.