Math
Equation of a Circle Calculator
The equation of a circle calculator moves between every form a class uses: center and radius → standard form → general form, general form back to center and radius by completing the square, and the unique circle through any three non-collinear points.
It also reports the radius exactly (√5, 5√2) and the circle's area and circumference in terms of π.
x-coordinate of the center.
y-coordinate of the center.
Must be positive.
Try an example
Result
Standard form
—
- General form
- —
- Center
- —
- Radius
- —
- Area
- —
- Circumference
- —
- x- and y-intercepts
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the equation of a circle calculator solves
The equation of a circle calculator moves between every form a class uses: center and radius → standard form → general form, general form back to center and radius by completing the square, and the unique circle through any three non-collinear points. It also reports the radius exactly (√5, 5√2) and the circle's area and circumference in terms of π.
Formula
How to use it
- Choose your starting point: center and radius, the general-form The number multiplying a variable, like the 3 in 3x. D, E, F, or three points.
- Enter the numbers (negative values are fine — keep the sign from the equation).
- Read the standard form, general form, center, exact radius, and A point where a graph crosses an axis.; open the work to see the completed square.
How to read the answer
Standard form shows the center and radius directly: (x − 2)² + (y + 3)² = 25 is centered at (2, −3) with radius 5 — note the sign flip inside the parentheses. General form hides them; completing the square recovers them. If r² comes out negative the equation has no real graph; if it is zero the 'circle' is a single point.
Common mistakes and edge cases
- Reading the center with the wrong signs: (x − 2)² means h = 2, and (y + 3)² means k = −3.
- Forgetting to take the square root — the right side of standard form is r², not r.
- When completing the square, adding (D/2)² to the left side but not to the right side.
- Three points on one line (collinear) do not lie on any circle; the calculator reports this instead of dividing by zero.
- A general form with a coefficient other than 1 on x² and y² must be divided through first; this calculator assumes x² + y² already have coefficient 1.
Worked examples
Center (2, −3), radius 5
Standard form
(x - 2)^2 + (y + 3)^2 = 25
General form x² + y² − 4x + 6y − 12 = 0
Standard form
(x - 2)^2 + (y + 3)^2 = 25
Through (1, 1), (2, 4), (5, 3)
Standard form
(x - 3)^2 + (y - 2)^2 = 5
Centered at the origin, radius 1 (unit circle)
Standard form
x^2 + y^2 = 1
Decimal center (−1.5, 0.5), radius 2.5
Standard form
(x + 1.5)^2 + (y - 0.5)^2 = 6.25
General form with no real points
Standard form
Error
Collinear points (no circle)
Standard form
Error
Frequently asked questions
What is the standard form of a circle's equation?+
(x − h)² + (y − k)² = r², where (h, k) is the center and r the radius. A circle centered at (2, −3) with radius 5 is (x − 2)² + (y + 3)² = 25.
How do I convert general form to standard form?+
Complete the square for x and for y. From x² + y² − 4x + 6y − 12 = 0: (x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9, so (x − 2)² + (y + 3)² = 25.
How do I find the center and radius from x² + y² + Dx + Ey + F = 0?+
Center = (−D/2, −E/2) and r² = (D/2)² + (E/2)² − F. If r² is negative there is no circle.
How do I find the equation of a circle through three points?+
Substitute each point into x² + y² + Dx + Ey + F = 0 to get three linear equations, solve for D, E, F, then complete the square. The points must not be on one line.
Why does the sign flip inside the parentheses?+
Standard form subtracts the center: (x − h). A center at x = −3 gives (x − (−3)) = (x + 3). So (x + 3)² means h = −3.
What does the calculator mean by exact radius like √5?+
The radius squared came out to 5, so the radius is exactly √5 ≈ 2.2361. Exact radicals are what algebra classes expect when r² is not a perfect square.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026