Equation of a Circle Calculator
This equation of a circle calculator uses (x − a)² + (y − b)² = r². With centre (2, −3) and radius 5 it gives (x − 2)² + (y + 3)² = 25. It converts between the forms and finds the circle through three points.
Example: Define the circle: from centre and radius · Centre x (a) 2 · Centre y (b) -3 · Radius r 5 · Test point x 5 · Test point y 1 → Equation of the circle: (x − 2)² + (y + 3)² = 25. Source: Wikipedia – Kreis (Kreisgleichung, Koordinatenform). Updated: .
How it is calculated
A circle is the set of all points at the same distance r from the centre M(a, b). The Pythagorean theorem gives the equation (x − a)² + (y − b)² = r². Expanding it gives the general form x² + y² + D·x + E·y + F = 0.
From one form to the other
From centre and radius it follows that D = −2a, E = −2b and F = a² + b² − r². Conversely, completing the square gives the centre M(−D/2, −E/2) and the radius r = √(D²/4 + E²/4 − F). For (x − 2)² + (y + 3)² = 25 the general form is x² + y² − 4x + 6y − 12 = 0. If the expression under the root is not positive there is no circle.
Circle through three points
Three points that do not lie on a line determine exactly one circle, the circumcircle of the triangle. Its centre is where the perpendicular bisectors meet. The calculator solves this directly from the coordinates and gives both forms. With a test point you can see whether a point lies inside, on or outside the circle by comparing its distance from the centre with r.
- A centre at the origin gives the simple form x² + y² = r².
- Coordinates are Cartesian; mind the signs: (y + 3) means b = −3.
- Circumference 2πr and area πr² are shown as well.
Frequently asked questions
What is the equation of a circle?
(x − a)² + (y − b)² = r² with centre M(a, b) and radius r. For M(0, 0) and r = 5 it is x² + y² = 25.
How do I find the centre and radius from the general form?
With M(−D/2, −E/2) and r = √(D²/4 + E²/4 − F). For x² + y² − 4x + 6y − 12 = 0 that gives M(2, −3) and r = 5.
How do you find the equation of a circle through three points?
The centre is where the perpendicular bisectors of the point pairs meet, and the radius is the distance to one point. The calculator does the work if the points are not on a line.
Sources and legal basis
- Wikipedia – Kreis (Kreisgleichung, Koordinatenform)
- Wikipedia – Circle (Equations)
- Wikipedia – Umkreis (circumcircle; calculation from the coordinates of the vertices)
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