The Equation of a Circle Calculator computes a circle’s equation from its center (h, k) and radius (r). It outputs both the standard form and the general form, so you can verify answers or graph the circle fast.
Circle equation basics (what the calculator uses)
A circle is the set of all points in a plane that are the same distance from a fixed point called the center. That common distance is the radius.
The calculator works with two common equation styles.
Standard form
The standard form of a circle is:
(x − h)2 + (y − k)2 = r2
- (h, k) is the center.
- r is the radius.
General form
The general form is:
x2 + y2 + Dx + Ey + F = 0
For a circle, D, E, and F come directly from h, k, and r after expanding the standard form.
Core formulas (the math behind the results)
Start from the standard form:
(x − h)2 + (y − k)2 = r2
Expand each square:
- (x − h)2 = x2 − 2hx + h2
- (y − k)2 = y2 − 2ky + k2
Combine terms and move r2 to the left side:
x2 + y2 − 2hx − 2ky + (h2 + k2 − r2) = 0
How the coefficients are computed
| General form term | Value |
|---|---|
| D (coefficient of x) | D = −2h |
| E (coefficient of y) | E = −2k |
| F (constant term) | F = h2 + k2 − r2 |
How to use the Equation of a Circle Calculator
Enter the circle’s center coordinates and radius. Then choose whether your inputs are in units (like meters or inches) or plain numbers.
The calculator returns:
- Standard equation: (x − h)2 + (y − k)2 = r2
- General equation: x2 + y2 + Dx + Ey + F = 0
- Computed coefficients D, E, and F
Common checks and interpretation
When you graph a circle, the equation should place the center at (h, k) and the distance to the circle at r.
- If r is larger, the circle is wider.
- If h or k changes, the circle shifts left/right or up/down.
- In the general form, the x and y linear coefficients reflect those shifts.
Use the calculator to confirm whether an equation you found by hand matches the given center and radius.
Practical examples
Example 1: Quick conversion for homework
You’re given a circle with center (3, −2) and radius 5. The standard form is directly:
(x − 3)2 + (y + 2)2 = 25
The calculator expands it to general form, producing D = −6, E = 4, and F = 9 + 4 − 25 = −12.
So the general equation is:
x2 + y2 − 6x + 4y − 12 = 0
Example 2: Engineering-style measurement in real units
Suppose a circular hole has center relative to a reference point at (h, k) = (0.12 m, 0.05 m) and radius r = 0.02 m. You can enter meters, then switch to inches if your work uses imperial units.
The calculator converts r (and keeps h, k consistent with the chosen unit system) and outputs both equation forms ready for graphing or verification.
Frequently Asked Questions
What is the standard form of the equation of a circle?
The standard form is (x − h)2 + (y − k)2 = r2. Here (h, k) is the center and r is the radius. This form is easiest to read because it shows the shifts and the exact radius directly.
How do you get the general form from the standard form?
Expand (x − h)2 and (y − k)2, then combine like terms. You will get x2 + y2 plus linear terms in x and y, and a constant term. Rearranging gives x2 + y2 + Dx + Ey + F = 0.
What do D, E, and F mean in the general equation of a circle?
In x2 + y2 + Dx + Ey + F = 0, D equals −2h and E equals −2k. The constant F equals h2 + k2 − r2. These values come from expanding the standard form.
Can the calculator handle negative centers and fractional radii?
Yes. Negative h or k simply shifts the center left or down. Fractional radii are also valid as long as r is greater than 0. The calculator validates inputs and will flag invalid radius values so you do not compute nonsensical equations.
How do I verify that an equation represents the correct circle?
Use the center (h, k) and radius r to test points. For example, plug the point (h + r, k) into the equation. If the left side equals 0 (for general form) or matches r2 (for standard form), the equation matches the intended circle.
Next steps
Use the results to graph the circle, check your work, or rewrite equations between standard and general forms. If you’re given a general equation, you can reverse-engineer h, k, and r using the same coefficients D and E.
When you need accuracy under time pressure, this Equation of a Circle Calculator gives the correct expanded equation in seconds.
