Equation of a Circle Calculator

The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly reveals the center and radius of the circle.

The general form is x² + y² + Dx + Ey + F = 0. It does not directly show the center or radius, but these can be recovered: center = (-D/2, -E/2) and radius = √(D²/4 + E²/4 - F).

Expand the squared terms: (x - h)² + (y - k)² = r² becomes x² - 2hx + h² + y² - 2ky + k² = r². Rearrange: x² + y² - 2hx - 2ky + (h² + k² - r²) = 0. So D = -2h, E = -2k, F = h² + k² - r².

Complete the square for x and y: x² + Dx + y² + Ey = -F. Group: (x² + Dx + D²/4) + (y² + Ey + E²/4) = -F + D²/4 + E²/4. This gives (x + D/2)² + (y + E/2)² = (D² + E² - 4F)/4. So h = -D/2, k = -E/2, r = √(D² + E² - 4F)/2.

Substitute each point into x² + y² + Dx + Ey + F = 0 to get three equations in three unknowns (D, E, F). Solve the system of linear equations. This uniquely determines the circle (three non-collinear points define a unique circle).

The center is the midpoint: h = (x₁ + x₂)/2, k = (y₁ + y₂)/2. The radius is half the distance: r = √((x₂-x₁)² + (y₂-y₁)²) / 2. Substitute into standard form.

No. Three collinear points lie on infinitely many circles (or none, depending on interpretation). The system of equations becomes inconsistent. The calculator will detect this and display an error.

For center (0, 0): x² + y² = r². This is the simplest form. For example, the unit circle has equation x² + y² = 1.

Complete the square to convert to standard form. The radius is r = √(D² + E² - 4F) / 2. If D² + E² - 4F < 0, the equation does not represent a real circle.

Substitute the point (x₀, y₀) into the equation. If (x₀ - h)² + (y₀ - k)² = r², the point is on the circle. If the left side is less than r², the point is inside. If greater, it is outside.

For a circle centered at the origin: r = a (constant radius). For a circle through the origin with center at (a, 0): r = 2a cos(θ). General polar forms involve both r and θ.

For a tangent at point (x₀, y₀) on the circle: (x₀ - h)(x - h) + (y₀ - k)(y - k) = r². The tangent is perpendicular to the radius at the point of tangency.

Yes. If D² + E² - 4F < 0, the equation has no real solutions (empty set). If D² + E² - 4F = 0, it represents a single point (degenerate circle with r = 0). Only when D² + E² - 4F > 0 does it represent a real circle.

If the circle passes through (0, 0), then F = 0 in the general form. The equation becomes x² + y² + Dx + Ey = 0, or in standard form: (x - h)² + (y - k)² = h² + k².

The circle equation is derived from the distance formula: the set of all points (x, y) at distance r from center (h, k) satisfies √((x-h)² + (y-k)²) = r, which simplifies to (x-h)² + (y-k)² = r².

The formula handles this naturally. For center (-3, 4): (x - (-3))² + (y - 4)² = r² becomes (x + 3)² + (y - 4)² = r². The sign convention always uses (x - h) and (y - k).

If tangent to both axes in the first quadrant with center (r, r): (x - r)² + (y - r)² = r². The center is equidistant from both axes, and the distance equals the radius.

For a chord at distance d from the center: chord length = 2√(r² - d²). If the chord passes through the center (d = 0), the chord is the diameter: length = 2r.

If the three points form an acute triangle, the smallest enclosing circle is the circumscribed circle. If the triangle is obtuse, the smallest enclosing circle has the longest side as diameter.

The radical axis is the line of points with equal power with respect to both circles. For circles x² + y² + D₁x + E₁y + F₁ = 0 and x² + y² + D₂x + E₂y + F₂ = 0, the radical axis is (D₁-D₂)x + (E₁-E₂)y + (F₁-F₂) = 0.

Circle equations are fundamental in coordinate geometry for finding intersections, tangents, chord properties, and area calculations. They are used in computer graphics, physics (orbital mechanics), engineering (gear design), and navigation (GPS trilateration).

The Equation of a Circle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Equation of a Circle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.