What Is the Standard Equation of a Circle?
A circle is the set of all points in a plane that are equidistant from a fixed central point. This distance is called the radius (r). The standard equation of a circle with center (h, k) and radius r is:
(x − h)² + (y − k)² = r²
Every point (x, y) that satisfies this equation lies on the circle. When the center is at the origin (0, 0), the equation simplifies to x² + y² = r².
Parametric Equation of a Circle
The same circle can be described parametrically using the angle parameter t (in radians):
x(t) = h + r · cos(t)y(t) = k + r · sin(t)- where t ∈ [0, 2π]
As t sweeps from 0 to 2π, the point (x(t), y(t)) traces the entire circle exactly once, starting and ending at (h + r, k).
Standard Equation to Parametric Equation
Given the standard form (x − h)² + (y − k)² = r², you can immediately write the parametric form by reading off h, k, and r. For example:
- Standard: (x − 3)² + (y + 4)² = 25 → h = 3, k = −4, r = 5
- Parametric: x(t) = 3 + 5·cos(t); y(t) = −4 + 5·sin(t)
How to Find the Equation of a Circle?
There are two common ways to determine the circle equation:
- From center and radius — substitute h, k, and r directly into the standard form. Example: center (2, −1), radius 7 → (x − 2)² + (y + 1)² = 49.
- From three points on the circle — any three non-collinear points uniquely define a circle. Substitute each point into the general form x² + y² + Dx + Ey + F = 0 to get a 3×3 linear system; solve for D, E, F and then convert back to standard form. This calculator performs all steps automatically.
Another Form of the Circle Equation (General Form)
Expanding the standard form and rearranging gives the general equation:
x² + y² + Dx + Ey + F = 0
where D = −2h, E = −2k, and F = h² + k² − r². Conversely, to convert from general to standard form, complete the square for both x and y:
- Group x and y terms: (x² + Dx) + (y² + Ey) = −F
- Complete the square: (x + D/2)² + (y + E/2)² = D²/4 + E²/4 − F
- Read center (h, k) = (−D/2, −E/2) and r = √(D²/4 + E²/4 − F)
How to Use This Equation of a Circle Calculator?
- Choose input mode: "Center & Radius" if you know the center coordinates and radius, or "Three Points on Circle" to determine the equation from three known points.
- Select a unit of length (metric or US imperial) to label circumference and area results.
- Enter the values for center (h, k) and radius, or for the three points.
- Click "Find Equation". The calculator instantly returns:
- Standard equation (h, k) form
- General (expanded) equation
- Parametric equations
- Center coordinates, radius, diameter, circumference, and area
- Use the Share button to copy a direct link to your result.
FAQs
What if the three points are collinear?
Collinear points (all on the same straight line) cannot define a circle — a circle requires three non-collinear points. The calculator will alert you if it detects this situation.
Can the center be outside the first quadrant?
Yes. The center (h, k) can have negative, zero, or positive coordinates. The formula works for any center position in the coordinate plane.
What units does this calculator support?
Metric: millimeters (mm), centimeters (cm), meters (m), kilometers (km).
US Imperial: inches (in), feet (ft), yards (yd), miles (mi).
The unit affects only the labels for radius, diameter, circumference, and area — the equation
itself is dimensionless.
How is the area of a circle calculated?
Area = π · r². For example, a circle with radius 5 cm has area = π · 25 ≈ 78.5398 cm².
What is the difference between standard and general form?
The standard form (x − h)² + (y − k)² = r² immediately reveals the center and radius. The general form x² + y² + Dx + Ey + F = 0 is obtained by expanding the standard form; it is useful for algebraic manipulation but requires completing the square to identify the center and radius.