Calculadora de geometria analítica do círculo

The Circle Analytic Geometry Calculator works with the algebraic equation of a circle on the Cartesian plane — not just its area and circumference, but where it sits, what its center and radius are, and how to express it in different equivalent forms. This tutorial covers the standard form (center–radius form), the general form, how to convert between them by completing the square, and how to recover the equation from given points.

The standard form of a circle

A circle with center (h, k) and radius r has the equation:

(x − h)² + (y − k)² = r²

This is called the standard form or center–radius form. The intuition: a point (x, y) is on the circle if and only if its distance from the center (h, k) equals r. By the distance formula, that distance is √((x − h)² + (y − k)²). Setting that distance equal to r and squaring both sides gives the standard form above.

Reading the equation:

• The signs flip: (x − h) means the center has x-coordinate +h, not −h. So (x − 3)² + (y − 5)² = 16 has center (3, 5), not (−3, −5).
• The right side is r², not r. The radius of (x − 3)² + (y − 5)² = 16 is √16 = 4.

Example — Build a standard-form equation

Given: center (2, −3), radius 5. Equation: (x − 2)² + (y − (−3))² = 5², which simplifies to (x − 2)² + (y + 3)² = 25.

The general form of a circle

If you expand the standard form (x − h)² + (y − k)² = r² and rearrange:

x² − 2hx + h² + y² − 2ky + k² = r²

x² + y² + (−2h)x + (−2k)y + (h² + k² − r²) = 0

Letting D = −2h, E = −2k, F = h² + k² − r², the equation becomes:

x² + y² + Dx + Ey + F = 0

This is the general form. Given D, E, F, you can recover the center and radius:

• h = −D/2
• k = −E/2
• r = √((D/2)² + (E/2)² − F) = √(D² + E² − 4F)/2

The radius formula requires the value under the square root to be positive: D² + E² > 4F. If it is exactly zero, the "circle" is a single point (degenerate). If it is negative, the equation has no real solution (an "imaginary circle").

Converting standard ↔ general

Standard → General: expand the squares and collect like terms.

Example: (x − 1)² + (y + 2)² = 9 → x² − 2x + 1 + y² + 4y + 4 = 9 → x² + y² − 2x + 4y − 4 = 0. So D = −2, E = 4, F = −4.

General → Standard: complete the square on x and y separately.

Example: x² + y² + 6x − 8y + 9 = 0.

• Group x and y terms: (x² + 6x) + (y² − 8y) = −9
• Complete the square: take half the coefficient, square it, add to both sides. Half of 6 is 3, 3² = 9. Half of −8 is −4, (−4)² = 16.
• (x² + 6x + 9) + (y² − 8y + 16) = −9 + 9 + 16
• (x + 3)² + (y − 4)² = 16

So this circle has center (−3, 4) and radius √16 = 4.

Finding the equation from given points

Case 1 — Center + a point on the circle. Given center (h, k) and any one point (x₀, y₀) on the circle, the radius is the distance from the center to that point: r = √((x₀ − h)² + (y₀ − k)²). Plug into standard form.

Case 2 — Three points on the circle. Any three non-collinear points determine a unique circle. Plug each point into the general form to get three equations in D, E, F:

x₁² + y₁² + Dx₁ + Ey₁ + F = 0
x₂² + y₂² + Dx₂ + Ey₂ + F = 0
x₃² + y₃² + Dx₃ + Ey₃ + F = 0

Three linear equations in three unknowns. Solve by elimination, substitution, or Cramer's rule. The AI Solve button on this calculator can walk through this for you — describe the three points and the AI sets up the system and solves it step-by-step.

Case 3 — Two endpoints of a diameter. The center is the midpoint of the two endpoints (use the midpoint formula), and the radius is half the distance between them.

What can go wrong

• Three collinear points do not define a circle — they define a line. The system of equations will be inconsistent or singular.
• Three identical points are not three points — they define infinitely many circles passing through that point.
• D² + E² < 4F in general form: no real circle exists. The equation has the algebraic form of a circle but no real (x, y) satisfies it.

Geometric meaning of the equation

The standard form makes immediate geometric sense: every circle is the set of points at a fixed distance from a fixed center. The general form is the same set of points written differently — algebraically convenient for some calculations (especially when you have a system that mixes circles and lines), geometrically opaque.

Two facts to internalize:

• The coefficient of x² and y² must be equal (and nonzero) for the equation to describe a circle. If they differ, you may have an ellipse, hyperbola, or parabola instead.
• There is no xy cross term in a circle equation. An xy term tilts the conic — you may have a rotated ellipse.

Common mistakes

• Sign flip on center. (x − 3)² means h = +3, not −3. Reading the standard form requires negating what is next to x and y.
• Forgetting to square-root the right side. If the equation says = 49, the radius is 7, not 49.
• Completing the square only halfway. You must (a) take half the coefficient, (b) square it, (c) add to both sides. Skipping step (c) breaks the equation.
• Treating the general form like a standard form. x² + y² + 4x − 6y = 12 is NOT (x + 4)² + (y − 6)² = 12. You have to complete the square first to extract the center.