Standard Equation Circle Calculator

Background.

Start with the simplest circle there is, x² + y² = r², centred on the origin. To slide it so its centre sits at (h, k), replace x with x − h and y with y − k. The result, (x − h)² + (y − k)² = r², is the standard form, and its minus signs are simply the cost of the shift: a centre two units right and three down gives (x − 2)² and (y + 3)², which looks backwards until you notice that x − 2 measures how far a point lies to the right of the centre.

In this form the circle can be read without any algebra. The centre sits in the brackets, the radius is the square root of the right-hand side, and the four extreme points, left, right, top and bottom, are one step away at (h ± r, k) and (h, k ± r). Where the circle crosses an axis comes out with a square root rather than the quadratic formula, because setting y = 0 or x = 0 leaves a single squared bracket to undo.

Give the page the centre coordinates h and k and a positive radius, and it returns the right-hand side, r², together with the constant F of the same circle multiplied out as x² + y² + Dx + Ey + F = 0, where F = h² + k² − r². The other two expanded coefficients need no calculation, D = −2h and E = −2k, so they are left for you to write in.

The radius must be greater than zero. A zero radius describes a single point, not a circle, and a negative one has no geometric meaning, so both are refused. Fitting a circle to measured points, or finding one through given points, is a different job from writing out a circle you already know.

What is standard equation circle calculator?

The standard equation of a circle, (x − h)² + (y − k)² = r², is the test a point (x, y) must pass to sit exactly r units from the centre (h, k). The left side is the square of that point's distance from the centre, built from its horizontal and vertical offsets; the right side is the square of the radius. Where the left side comes out smaller the point is inside the circle, and where it comes out larger the point is outside.

How to use this calculator.

1. 01Enter the centre's x-coordinate as h and its y-coordinate as k, each with its own sign. A centre at (2, −3) goes in as 2 and −3; the sign flip that appears inside the brackets is something you write, not something you enter.
2. 02Enter the radius as a positive number. If you were given a diameter, halve it first; if you were given r² from an existing equation, take its square root.
3. 03Write the standard form from h, k and the r² shown. For the expanded form, use D = −2h, E = −2k and the F returned here in x² + y² + Dx + Ey + F = 0.
4. 04Check by substituting a point you know is on the circle, such as (h + r, k), into either form; both should balance exactly.

The formula.

The standard form is (x − h)² + (y − k)² = r², so the right-hand side is just the radius squared. Multiplying out the brackets gives x² − 2hx + h² + y² − 2ky + k² = r², and gathering everything on one side gives x² + y² − 2hx − 2ky + (h² + k² − r²) = 0. The constant term F = h² + k² − r² is the second number the page returns. Because F folds the centre and radius together, it grows quickly as the centre moves away from the origin: a radius-5 circle centred at (100, 0) has F = 9,975, while the same circle at (2, −3) has F = −12. F is the squared distance from the origin to the centre minus r², so its sign places the origin: inside the circle when F is negative, on it when F is zero, outside when F is positive.

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.