A circle on the coordinate plane is every point at a fixed distance r from a center (h, k). Its equation comes in two common shapes — the standard form, which shows the center and radius at a glance, and the expanded general form. This calculator builds both from whatever information you have: center and radius, center and a point, the ends of a diameter, three points on the circle, or an equation in general form. All arithmetic is exact, and the circle is drawn on a grid.
How to use the circle equation calculator
- Choose what you know: center and radius, center and a point on the circle, the endpoints of a diameter, three points, or the general form coefficients D, E, F.
- Enter the numbers; decimals are converted to exact fractions.
- Read the standard form on the tape, with the general form, center, radius (in simplest radical form when needed), diameter, circumference, area and intercepts. The graph plots the circle, its center and your points.
Circle equation formulas
Going from general to standard form, the center is (−D/2, −E/2) and r2 = h2 + k2 − F.
| What you know | How the center and radius are found |
|---|---|
| Center and a point | radius = distance from the center to the point |
| Diameter endpoints | center = midpoint; radius = half the endpoints’ distance |
| Three points | solve for D, E, F from three linear equations |
| General form | complete the square in x and y |
Worked examples
Three points. Find the circle through (1, 1), (7, 1) and (1, 9). Substituting into x2 + y2 + Dx + Ey + F = 0 gives D + E + F = −2, 7D + E + F = −50 and D + 9E + F = −82. Solving: D = −8, E = −10, F = 16.
The center is (4, 5) and r2 = 16 + 25 − 16 = 25, so the circle is (x − 4)2 + (y − 5)2 = 25, radius 5. Check (7, 1): 9 + 16 = 25 ✓.
Center and radius. Center (2, −3), radius 5: (x − 2)2 + (y + 3)2 = 25. Expanding gives x2 + y2 − 4x + 6y − 12 = 0.
The three given points form a right angle at (1, 1), so the segment from (7, 1) to (1, 9) is a diameter — the circle’s center is its midpoint (4, 5) and its length 10 is twice the radius. Thales’ theorem guarantees this for every right triangle.
Reading intercepts
Setting y = 0 in the standard form gives (x − h)2 = r2 − k2:
- if r2 − k2 > 0, the circle crosses the x-axis twice;
- if it equals 0, the circle just touches the axis (tangent), as the example circle does at (4, 0);
- if it is negative, the circle misses the axis.
The y-intercepts work the same way with the roles of x and y swapped.
Common mistakes
- Sign of the center. (x + 3)2 means h = −3, not 3.
- Radius versus radius squared. The right side of the standard form is r2; a right side of 20 means r = √20 = 2√5.
- Unequal x² and y² coefficients. If they differ, the curve is an ellipse, not a circle. If they are equal but not 1, divide the whole equation by that coefficient first.
Related tools
For area and circumference from a radius, use the circle calculator. The distance calculator and midpoint calculator cover the building blocks, the completing the square calculator explains the conversion from general form, and the circumcircle and incircle calculator finds the circle through a triangle’s corners from its side lengths.
Frequently asked questions
What is the standard form of a circle's equation?
(x − h)² + (y − k)² = r², where (h, k) is the center and r the radius. A circle centered at (2, −3) with radius 5 is (x − 2)² + (y + 3)² = 25.
How do I convert the general form to standard form?
Complete the square in x and in y. For x² + y² − 4x + 6y − 12 = 0, group (x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9, giving (x − 2)² + (y + 3)² = 25.
How is a circle found from three points?
Substitute each point into x² + y² + Dx + Ey + F = 0 to get three linear equations in D, E and F, then solve them. The points must not lie on one line, or no circle passes through all three.
When does the general form not describe a circle?
After completing the square, the right side must be positive. If it is zero, the equation describes a single point; if it is negative, no real point satisfies it.