Equation of a Circle Calculator
Generate circle equations in standard and general forms instantly from center coordinates and radius.
Enter the center coordinates (h, k) and radius r to get both the standard form (x−h)² + (y−k)² = r² and the expanded general form, plus area and circumference.
Equation of a Circle Calculator
Generate circle equations in standard and general forms instantly from center coordinates and radius.
About the equation of a circle calculator
A circle is defined as the set of all points in a plane that are equidistant from a fixed center point. The constant distance from the center to any point on the circle is called the radius. This geometric definition translates directly into an algebraic equation that describes the circle with complete precision.
The standard form of a circle equation is (x − h)² + (y − k)² = r², where (h, k) is the center of the circle and r is its radius. This form derives directly from the distance formula: the distance between any point (x, y) on the circle and the center (h, k) is √[(x − h)² + (y − k)²], and setting that equal to r and squaring both sides yields the standard form. The great advantage of the standard form is that it makes the center and radius immediately visible without any algebra.
The general form of a circle equation is x² + y² + Dx + Ey + F = 0. It is obtained by expanding the standard form and collecting all terms on one side. The coefficients relate to the center and radius as follows: D = −2h, E = −2k, and F = h² + k² − r². The general form is useful for algebraic manipulation, solving systems of equations involving circles, and applications in calculus such as finding areas bounded by curves.
Converting between forms is a fundamental skill. To go from standard to general form, expand the squared binomials and rearrange. To go from general form back to standard form, complete the square on both the x-terms and the y-terms. Completing the square means rewriting x² + Dx as (x + D/2)² − (D/2)², which isolates the center coordinate as −D/2.
The area of a circle is A = πr², and the circumference is C = 2πr. Both depend only on the radius, so once the equation is known the geometric measurements follow immediately. For a unit circle centered at the origin, r = 1, so A = π and C = 2π — the simplest and most studied circle in mathematics.
Circle equations have broad practical applications. In computer graphics and game development, they are used for collision detection: two circles with centers (h₁, k₁) and (h₂, k₂) and radii r₁ and r₂ overlap when the distance between their centers is less than r₁ + r₂. In engineering, circular cross-sections of pipes, gears, and wheels are described by circle equations for tolerance and fit calculations. In astronomy, simplified circular orbits provide first-order approximations before refining to ellipses.
Understanding sign conventions is critical. In the standard form (x − h)² + (y − k)², the center x-coordinate h appears with a minus sign. So a center at (3, −2) gives (x − 3)² + (y − (−2))² = (x − 3)² + (y + 2)² = r². Students frequently make sign errors here, writing (x + 3)² instead of (x − 3)². The calculator automatically handles these conventions, displaying the equation in fully simplified, human-readable notation.
Circle equation examples
Four representative cases showing different center and radius configurations.
| Center & Radius | Standard Form | Note |
|---|---|---|
| Center (0, 0), r = 1 | x² + y² = 1 | The unit circle centered at the origin — the most fundamental circle in trigonometry. |
| Center (3, 4), r = 5 | (x − 3)² + (y − 4)² = 25 | A classic Pythagorean-triple circle; area = 25π ≈ 78.54, circumference = 10π ≈ 31.42. |
| Center (−2, −3), r = 6 | (x + 2)² + (y + 3)² = 36 | Circle in the third quadrant; note how negative center coordinates become positive signs in the equation. |
| Center (1.5, −2.5), r = 7.5 | (x − 1.5)² + (y + 2.5)² = 56.25 | Decimal inputs work seamlessly; area = 56.25π ≈ 176.71 square units. |
How to use the circle equation calculator
- Enter the x-coordinate of the center (h) — this can be any real number including negatives, decimals, or zero.
- Enter the y-coordinate of the center (k) — same rules apply.
- Enter the radius r as a positive number greater than zero. Decimal values are accepted for precision work.
- Click Calculate Equation to see the standard form, general form, area, and circumference instantly.
- Click Reset to clear all fields and start a new calculation.
Circle equation FAQ
What is the standard form of a circle equation?
The standard form is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. This form is derived from the distance formula and makes the geometric properties of the circle immediately readable without any further algebra.
How do I convert from standard form to general form?
Expand the squared binomials: (x − h)² + (y − k)² = r² becomes x² − 2hx + h² + y² − 2ky + k² = r². Move all terms to one side to get x² + y² − 2hx − 2ky + (h² + k² − r²) = 0, which is the general form x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k, and F = h² + k² − r².
What happens if the center is at the origin?
When h = 0 and k = 0, the standard form simplifies to x² + y² = r². The terms (x − 0)² and (y − 0)² reduce to x² and y², so the equation is much cleaner. For example, a circle centered at the origin with radius 5 has equation x² + y² = 25.
Can the radius be negative or zero?
No. A negative radius has no geometric meaning because the radius represents a distance, which is always non-negative. A radius of zero would reduce the circle to a single point, which is a degenerate case and not a true circle. The calculator requires a positive radius.
How is the circle equation used in collision detection?
In game physics and graphics, two circles with centers (h₁, k₁) and (h₂, k₂) and radii r₁ and r₂ are colliding when the Euclidean distance between their centers is less than or equal to r₁ + r₂. Computing this distance as √[(h₂ − h₁)² + (k₂ − k₁)²] and comparing it to the sum of radii is an efficient O(1) test for overlap.
How do I find the center and radius from a general form equation?
From x² + y² + Dx + Ey + F = 0, complete the square on x and y: h = −D/2, k = −E/2, and r = √[(D² + E² − 4F)/4]. For example, x² + y² + 6x − 8y + 15 = 0 gives h = −3, k = 4, and r = √[(36 + 64 − 60)/4] = √10 ≈ 3.162.