Circle from 3 Points Calculator
This circle from 3 points calculator takes three coordinates and returns the equation of the circle through them — in both forms, with the centre kept as an exact fraction rather than a rounded decimal.
Type (0, 0), (4, 0) and (0, 3) and you get x² + y² − 4x − 3y = 0, centre (2, 3/2), radius 5/2.
What the Circle from 3 Points Calculator Does
Any three points that are not on a straight line have exactly one circle through them. Finding it is a three-by-three simultaneous equation, and this page does that in exact fractions and then puts all three points back to check.
Start From the General Form
The standard form buries the centre inside brackets, which makes it awkward to substitute into. The general form leaves the three unknowns sitting on their own:
x² + y² + Dx + Ey + F = 0
Put a point in and the squares become plain numbers, so what is left is one ordinary linear equation:
x₁D + y₁E + F = −(x₁² + y₁²)
Three points give three of those. Solve them however you like — elimination, substitution, determinants — and nothing about the circle makes it harder than any other three-by-three system.
Check for a Straight Line First
This is the single commonest mistake in the topic, and it costs people whole questions: they solve first and then get lost when the arithmetic collapses.
x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) = 0 → on a line
That expression is twice the signed area of the triangle the points make. Zero area means no triangle, and no circle passes through three points on a line — two of them would sit on it and the third would fall inside or outside.
The Same Number Does Both Jobs
Here is the part worth seeing rather than memorising. When you solve the system with determinants, the number you divide by is that same expression.
So the calculation refuses to work in exactly the case where no answer exists. The algebra and the geometry agree without being made to — you do not have to remember to check for a line, because the check is already inside the solve.
Expect a Fraction, and Keep It
A circle through three whole-number points hardly ever has a whole-number centre. Take (0, 0), (1, 3), (4, 2) — the centre lands on (2, 1) and the radius is √5. Move one point slightly and you are into sixths and elevenths.
Round 11/6 to 1.833 and everything after it is slightly wrong: the radius, and whether a fourth point is really on the circle. This page carries the fractions to the end, and gives the radius as a surd when it is not whole.
Three Points Almost on a Line
Take (0, 0), (1, 1) and (2, 2.001). They very nearly form a line, so the circle through them is enormous — and a page working in decimals will tell you the answer is unstable or undefined, because rounding errors swamp the calculation.
There is nothing unstable about it. In exact fractions there is no "nearly": the determinant is zero or it is not, and here it is not. The centre comes out at (−4002001/2000, 4004001/2000), all three points sit on the circle exactly, and the page hands it to you without complaint.
Two Points Are Not Three
If you type the same point twice, you have given two points rather than three — and infinitely many circles pass through two points. That is a different mistake from collinearity and is told apart here, because being sent to look for a straight line when what you did was repeat a coordinate wastes the time the page was meant to save.
Using the Circle from 3 Points Calculator
Fill in the six coordinates. Whole numbers, fractions like 1/2 and decimals all work, and a minus sign is fine. Tap a box to send the keypad there.
You get the equation in both forms, the centre and radius, the diameter, circumference and area, and a drawing of the three points with the circle through them — or, when they are on a line, the three points sitting on that line and no circle at all.
Circle from 3 Points Calculator FAQ
How do you find the equation of a circle through three points?
Can a circle pass through any three points?
How do you check whether three points are collinear?
Why is the centre a fraction?
Check for a line, solve the three equations, and put all three points back before you write the answer down.
