The circle equation calculator above moves a circle between the two ways it is normally written. Standard form, (x − h)² + (y − k)² = r², shows you the centre and the radius directly. General form, x² + y² + Dx + Ey + F = 0, hides both behind expanded brackets and is what you usually get handed in a textbook question or produced by an algebra step. The tool converts either way, and it will also assemble the equation from a centre and radius, from two endpoints of a diameter, or from a centre and any single point known to sit on the circle.
Arb Digital publishes free calculators for the arithmetic people redo by hand and get wrong in the same three places every time. Circle equations are a clear example: the sign inside the bracket flips, the constant gets added when it should be subtracted, and the halving step in completing the square lands on the wrong number. Showing the centre, the radius, both forms and the derived circumference and area side by side makes a slip visible immediately rather than three lines later.
What This Circle Equation Calculator Does
It works entirely in coordinate geometry. Give it four coefficients and it completes the square on both variables to recover the centre and radius. Give it a centre and radius and it expands the brackets back out to general form. Give it two points and it treats them either as the ends of a diameter, taking the midpoint as centre, or as a centre plus a point on the rim, taking their separation as the radius.
That is a different job from our circle calculator, which measures a circle: give it a radius and it returns diameter, circumference, area and arc lengths, with no coordinate plane involved at all. This page never asks where a circle is; that one never asks how big it is. It is also distinct from the unit circle calculator, which is a trigonometry reference for sine and cosine values at standard angles on the fixed circle of radius one centred at the origin. If you want the equation of a circle, you are in the right place. If you want its measurements or its trigonometry, follow those links instead.
How to Use It
- Pick what you already have. The mode selector changes which fields appear. Everything else on the page updates the moment you type, so there is no need to press the button unless you want to.
- Enter the coefficients with their signs. In general form, move every term to the left of the equals sign first. An equation written as x² + y² = 25 has D = 0, E = 0 and F = −25, not +25.
- Check the leading coefficient. If the equation reads 2x² + 2y² + 8x − 4 = 0, put 2 in the A field. The tool divides through for you rather than silently assuming a monic equation.
- Read the headline equation. The hero line is the standard form with the signs already resolved, so a centre at y = −2 is printed as (y + 2)² and not as (y − −2)².
- Use the supporting grid as a sanity check. Centre, radius, circumference and area all derive from the same two numbers. If the radius looks wrong, the area will look obviously wrong too.
The Formula and How It's Calculated
A circle is the set of points at a fixed distance from a fixed centre. Applying the distance formula to that sentence gives standard form directly: the point (x, y) lies on the circle when √((x − h)² + (y − k)²) = r, and squaring both sides removes the root to leave (x − h)² + (y − k)² = r². Expanding those brackets produces x² − 2hx + h² + y² − 2ky + k² − r² = 0, which is general form with D = −2h, E = −2k and F = h² + k² − r².
Reversing that gives the conversion the calculator performs most often. From the three relationships above, h = −D/2, k = −E/2, and r² = h² + k² − F. Take the default values of D = −6, E = 4 and F = −12. Then h = 6/2 = 3, k = −4/2 = −2, and r² = 9 + 4 + 12 = 25, so r = 5. The standard form is (x − 3)² + (y + 2)² = 25, the circumference is 2π(5) = 31.4159, and the area is π(25) = 78.5398. Every one of those numbers appears on the page above, which makes this worked example easy to verify by hand. Wolfram MathWorld's entry on the circle derives the same centre and radius formulas from the general quadratic curve.
Completing the Square, Step by Step
The formulas above are shortcuts for a process worth being able to do by hand, because exams ask for the working and because the same technique appears again in conic sections and in optimisation. Start from x² + y² − 6x + 4y − 12 = 0 and group the variables: (x² − 6x) + (y² + 4y) = 12. Notice the constant has already moved to the right and changed sign.
Now handle each bracket separately. For the x group, halve the coefficient of x to get −3, square it to get 9, and add 9 inside the bracket. Because you added 9 to the left, you must add 9 to the right as well. For the y group, half of 4 is 2, and 2 squared is 4, so add 4 to both sides. The equation becomes (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4, which factors into (x − 3)² + (y + 2)² = 25.
Two errors dominate here. The first is forgetting to add the same amount to the right-hand side, which changes the radius but not the centre and produces a plausible-looking wrong answer. The second is halving the coefficient of x² rather than the coefficient of x. Lamar University's algebra notes on circles work through the same procedure with graphed examples.
When the General Form Is Not a Circle at All
Nothing stops you writing down an equation of the form x² + y² + Dx + Ey + F = 0 that no circle satisfies. The test is the sign of r² = h² + k² − F. If that quantity is positive you have a genuine circle. If it is exactly zero the equation is satisfied by the single point (h, k) — a degenerate circle of radius zero, sometimes called a point circle. If it is negative, no real point satisfies the equation and the graph is empty; the solutions exist only in the complex plane, which is why textbooks sometimes call it an imaginary circle.
Try it. Leave D and E at their defaults and change F to 20. Then r² = 9 + 4 − 20 = −7, and the calculator says no real circle exists rather than returning NaN. Set F to 13 instead and r² = 0, giving the point circle at (3, −2). A degenerate case is genuine information about the equation, and it usually means a sign was flipped upstream.
There is a second precondition that is easy to miss. The general form only describes a circle when the x² and y² coefficients are equal and there is no xy term. Unequal coefficients give an ellipse, opposite signs give a hyperbola, and a non-zero xy term gives a conic rotated off the axes. This calculator asks for one shared coefficient A precisely so that the assumption is stated rather than hidden. For general second-degree equations in one variable, our quadratic equation solver is the right tool.
Building a Circle From Two Diameter Endpoints
The diameter mode solves a question that comes up constantly in coordinate geometry: two points are given as opposite ends of a diameter, and you need the equation. The centre is the midpoint, ((x₁ + x₂)/2, (y₁ + y₂)/2), and the radius is half the distance between them. With the default second-mode values of (0, 1) and (6, −5), the midpoint is (3, −2) and the separation is √(36 + 36) = 8.4853, so the radius is 4.2426 and r² is exactly 18.
There is a neat alternative. Because the angle in a semicircle is a right angle, any point (x, y) on the circle makes the vectors to the two endpoints perpendicular, so their dot product is zero. That gives the equation directly as (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0, with no midpoint or square root needed, and it expands to the same general form. If you want to see that perpendicularity test on its own, our dot product calculator reports whether two vectors are orthogonal, and the 3D distance calculator extends the underlying distance formula to a third coordinate, where the same algebra describes a sphere.
Where Circle Equations Are Used Outside Algebra Class
The centre-and-radius form is the standard description of a circular constraint in any coordinate system. Collision detection tests whether the distance between two centres is less than the sum of two radii, and it does so without ever taking a square root, because comparing squared distances is faster and avoids floating-point error. Geofencing works the same way: a radius around a point, and a distance test on every incoming location.
Trilateration, the arithmetic behind satellite positioning and indoor Wi-Fi location, intersects three circles whose radii come from measured distances. The intersection point is found by subtracting one circle's general form from another's, which cancels the x² and y² terms and leaves a straight line — the radical line of the two circles. Two such lines intersect at the position. That cancellation is the practical reason general form is worth keeping around at all, and it is why our line intersection calculator is a natural next step. When circular positions are given as an angle and a distance rather than as x and y, the polar coordinates converter translates between the two conventions.
Arb Digital builds measurement and reporting that shows its working, the same way this page shows every intermediate value instead of a single unexplained answer.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Reading h and k straight off the brackets with the wrong sign — (x + 4)² means h = −4, because standard form subtracts h. The sign you see is the opposite of the coordinate.
- Taking r from the right-hand side without a square root — the equation ends in r², so (x − 1)² + y² = 9 has radius 3, not 9.
- Leaving a constant on the right in general form — F is the constant after everything has been collected on one side. Reading it from x² + y² − 6x = 12 gives F = −12, not 12.
- Ignoring a leading coefficient — 3x² + 3y² + 6x = 0 must be divided through by 3 before the h = −D/2 shortcut applies, or the centre comes out three times too far from the origin.
- Adding to one side only when completing the square — the compensating addition on the right is what keeps the equation true, and skipping it changes the radius while leaving the centre correct, which makes the error hard to spot.
Related Free Tools From Arb Digital
Measure a circle once you know its radius with the circle calculator, look up exact trigonometric values with the unit circle calculator, work distances across the earth's surface with the great circle distance calculator, switch between coordinate systems with the coordinates converter, or find where two straight lines meet with the line intersection calculator. The full free online tools hub lists every mathematics tool we publish.
Frequently Asked Questions
It is (x minus h) squared plus (y minus k) squared equals r squared, where (h, k) is the centre and r is the radius. Standard form is preferred because both facts about the circle can be read directly from it.
Halve each of the x and y coefficients and change their signs. For x squared plus y squared plus Dx plus Ey plus F equals zero, the centre is at h equals minus D over 2 and k equals minus E over 2.
Compute r squared as h squared plus k squared minus F using the centre you just found, then take the square root. If that quantity is negative, the equation has no real circle.
Because standard form subtracts the centre coordinate. A circle centred at x equals minus 2 is written as (x plus 2) squared, since subtracting a negative two adds two.
Yes. When h squared plus k squared minus F comes out negative, no real point satisfies the equation and the graph is empty. When it is exactly zero the graph is a single point at the centre.
Then the curve is not a circle. Equal coefficients with the same sign give a circle, unequal coefficients give an ellipse, and opposite signs give a hyperbola.
The centre is the midpoint of the two points and the radius is half the distance between them. Switch the mode selector to the diameter option and the calculator does both steps.
