Equation of a Circle Calculator

Enter a circle's center coordinates and radius to get its equation in standard (center-radius) form and general (expanded) form, plus area and circumference.

Quick Facts

Standard form
(x - h)² + (y - k)² = r²
Center (h, k), radius r — every point (x, y) on the circle is exactly r units from the center.
General form
x² + y² + Dx + Ey + F = 0
Expanding standard form gives D = -2h, E = -2k, F = h² + k² - r². Both forms describe the same circle.

Your Results

Calculated
Standard form
-
(x - h)² + (y - k)² = r²
General form
-
x² + y² + Dx + Ey + F = 0
Area
-
π × r² (square units)
Circumference
-
2 × π × r (units)

Ready

Enter the center coordinates and radius, then press Calculate.

Formula and method for the Equation of a Circle

The equation of a circle describes every point (x, y) that sits exactly one fixed distance — the radius — from one fixed point, the center. This calculator takes a center (h, k) and a radius r and returns the equation in both standard and general form, along with the circle's area and circumference.

The standard (center-radius) form

The standard form of a circle's equation is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. It comes directly from the distance formula: if (x, y) is any point on the circle, its distance to the center (h, k) must equal r, and squaring that distance equation produces the standard form.

The general (expanded) form

Expanding the two squared terms and collecting everything on one side gives the general form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, and F = h² + k² − r². Both forms describe the exact same circle — standard form makes the center and radius obvious at a glance, while general form is how a circle's equation often appears after algebra or inside other formulas.

Area and circumference

Once the radius is set, two more measurements follow directly from it: area = π × r² (the space enclosed by the circle) and circumference = 2 × π × r (the distance around it). Both use the same radius that defines the equation, so a larger radius grows the area faster (by the square) than it grows the circumference (linearly).

Common mistakes

  • Sign errors: standard form subtracts h and k, so a center at (−3, 4) produces (x + 3)² + (y − 4)² = r², not (x − 3)² + (y + 4)² = r².
  • Radius vs. r²: the right side of standard form is r², not r. A circle of radius 5 has "= 25" on the right, not "= 5".
  • Zero or negative radius: a radius must be a positive number. r = 0 collapses the circle to a single point, and a negative radius has no geometric meaning.

Real-world applications

  • Computer graphics and game development use the circle equation for collision detection — checking whether a point falls inside a circular hit region
  • Mapping and location apps use it to define circular geofences and search radii around a point
  • Engineering and manufacturing use it to specify circular parts, pipes, gears, and tolerances
  • Analytic geometry courses use it as the entry point into conic sections and coordinate-plane proofs

Frequently Asked Questions

What is the standard form of a circle's equation?
The standard (center-radius) form is (x − h)² + (y − k)² = r², where (h, k) is the circle's center and r is its radius. Any point (x, y) that satisfies this equation lies exactly r units from the center.
How do I convert standard form to general form?
Expand the squares: (x − h)² + (y − k)² = r² becomes x² − 2hx + h² + y² − 2ky + k² = r². Move every term to one side to get x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, and F = h² + k² − r².
Can the radius be zero or negative?
No. A radius of zero shrinks the circle to a single point at the center, and a negative radius has no geometric meaning, so it cannot appear in a real circle's equation. This calculator requires a radius greater than 0.
How do I find the radius from area or circumference?
Solve the area formula for r: r = √(Area / π). From circumference, r = Circumference / (2π). Either result can then be substituted back into the standard-form equation to get the full equation of the circle.

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