Circle Equation Calculator
This circle equation calculator takes the whole equation as you have it written and gives the centre and radius — with the radius left exact rather than rounded, and a name for the shape when it turns out not to be a circle.
Type x² + y² − 6x + 4y − 12 = 0 and you get centre (3, −2), radius 5.
What the Circle Equation Calculator Does
Most tools for this ask you to fill in D, E and F in separate boxes. That sounds helpful and is not: picking those three numbers out of the equation is half the exercise, and it quietly refuses anything that does not already start with a bare x² + y².
Here you paste the line from your book. General form, standard form, terms on both sides of the equals sign, a shared coefficient in front of both squares — all of it goes in as written.
Two Forms, and Only One Tells You Anything
(x − h)² + (y − k)² = r² against x² + y² + Dx + Ey + F = 0
The first hands you the centre (h, k) and the radius by eye. The second hides them. The whole of this topic is turning the second into the first, and there is exactly one tool for the job: completing the square, done once for the x terms and once for the y terms.
One Worked All the Way Through
Take x² + y² − 6x + 4y − 12 = 0. Group the x terms and the y terms, and move the number across:
(x² − 6x) + (y² + 4y) = 12
Now complete each square. Halve the x coefficient: half of −6 is −3, and −3 squared is 9. Halve the y coefficient: half of 4 is 2, and 2 squared is 4. Add both to each side — that is the step people forget:
(x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4
(x − 3)² + (y + 2)² = 25
Centre (3, −2), radius 5. Notice the two signs disagree with the brackets on purpose — more on that below.
Divide Before Anything Else
Completing the square only works when the squared terms carry a coefficient of 1. If they carry anything else, divide the whole equation through by it first:
4x² + 4y² − 16x − 24y + 51 = 0 → divide every term by 4 → centre (2, 3), radius ½
Skipping this is the commonest way an answer goes wrong while every later step still looks right. And note what the coefficients tell you before you start: they must be equal. If the x² and y² terms carry different numbers, no amount of algebra will make a circle out of it.
Where the Standard Form Comes From
The form is not arbitrary, and seeing why makes it stop being something to memorise. A circle is every point that sits a fixed distance r from a fixed point (h, k). Write that distance out:
√[ (x − h)² + (y − k)² ] = r
Square both sides and the root disappears, leaving the standard form exactly as it is written. That is the whole derivation. It also explains why the number on the right is r² rather than r — the squaring is what put it there.
The Sign Flip That Costs Marks
The standard form is written with a minus inside each bracket, so a plus sign there means the coordinate is negative:
(x + 2)² + (y − 3)² = 16 is centred at (−2, 3)
More marks go here than on the algebra itself. The cure is small: write the bracket out once as (x − (−2)) before you read it off, and the sign stops catching you.
The Radius Is Usually Irrational
The number on the right of the standard form is r², not r — and once you take the square root, expect it not to be tidy. Most circles do not have a whole-number radius:
x² + y² − 2x − 2 = 0 → centre (1, 0), radius √3
√3 is the answer. 1.732 is a rounded copy of it, and most calculators hand you the copy. This one keeps the surd, in lowest form, so x² + y² = 12 gives 2√3 rather than √12.
There is a small reward for working this way. The area is πr², and r² is a whole number even when r is not — so the circle with radius √3 has an area of exactly 3π, which is tidier than its own radius. The working here gives the diameter, circumference and area alongside, all exact.
Three Things It Can Turn Out to Be
Completing the square does not promise you a circle. Look at what lands on the right:
r² > 0 — a circle.
r² = 0 — only the centre satisfies the equation. x² + y² − 4x − 6y + 13 = 0 is the single point (2, 3), not a circle of radius nothing.
r² < 0 — no point in the plane satisfies it, because two squares added together are never negative. x² + y² + 4 = 0 is one of these.
All three are real answers. Writing "no solution" for the last two throws away the part the question was actually testing.
When It Is Not a Circle at All
If the two squared coefficients differ, or there is an xy term, you have something else — and knowing what is usually the next thing you need. This page says so instead of stopping at "not a circle":
x² + 2y² = 4 is an ellipse. x² − y² = 4 is a hyperbola. y = x² is a parabola.
And some equations that look like conics have collapsed. x² − y² = 0 is not a hyperbola — it is two straight lines crossing. x² − 4 = 0 is a pair of parallel lines. These are told apart properly here, not lumped in with the curve they resemble.
Going the Other Way
Exam questions run this in reverse as often as forwards: given a centre and a radius, write the equation. That direction needs no square-completing at all — you already have the standard form, and expanding it gives the general one.
Centre (3, −2), radius 5: (x − 3)² + (y + 2)² = 25
Multiply out and collect, and you are back at x² + y² − 6x + 4y − 12 = 0. The two forms are the same equation wearing different clothes, which is worth proving to yourself once by doing exactly this and watching it close the loop.
A related question gives you the centre and a point on the circle instead of the radius. Find the distance between them — that distance is the radius — and then you are back to the case above. Do not round it on the way: if the distance comes out √13, the equation ends in = 13, and rounding first would have cost you the exact answer.
Is a Point On, Inside or Outside?
Once you have the centre and the radius, a question that looks like a new topic turns out to be one subtraction. Compare the squared distance from the centre against r² — there is no need to take any square roots at all:
For x² + y² − 6x + 4y − 12 = 0, centre (3, −2) and r² = 25:
(6, 2): 3² + 4² = 25 — equal to r², so the point is on the circle.
(0, 0): 3² + 2² = 13 — less than 25, so inside.
(10, 0): 7² + 2² = 53 — more than 25, so outside.
Staying with squares is the trick worth keeping. The moment you take a root you are usually holding an irrational number, and comparing two of those by hand is where errors creep in that the comparison never needed.
Using the Circle Equation Calculator
Type the equation on one line. Use ^ for a power — x^2, not x2, since the second could equally mean x times 2 and this page will not guess between them. Brackets go where you would write them, and both x and y are on the keypad along with the equals sign.
Every step is shown: the division if one was needed, the two squares completed, the standard form, and the circle drawn to scale from the numbers. The drawing changes with the answer, so a single point looks like a single point and an impossible equation draws nothing at all.
Circle Equation Calculator FAQ
How do you find the centre and radius from the general equation of a circle?
Why does (x + 2) squared mean the centre is at minus 2?
What if the radius squared comes out negative?
Do the x squared and y squared coefficients have to be 1?
Divide first, complete both squares, then read the centre off — and take the square root last, expecting a surd.