Equation of a Circle Calculator (center-radius & general form)

Understanding the circle equation calculator — concept and principle

A circle is the set of all points at the same distance, the radius r, from a fixed centre (h, k). This calculator writes the circle's equation in two forms: the standard (centre-radius) form, which shows the centre and radius at a glance, and the general form, which is the expanded polynomial often found in textbooks, graphing software and exam questions. Converting between the two is a routine step in analytic geometry.

How it is calculated

• Standard form: (x − h)² + (y − k)² = r².
• General form: x² + y² + Dx + Ey + F = 0, with D = −2h, E = −2k and F = h² + k² − r².
• Back from general to standard: centre (−D/2, −E/2), radius r = √(D²/4 + E²/4 − F).

r must be positive; h and k can be any real numbers.

The principle

Take any point (x, y) on the circle. Its horizontal distance from the centre is x − h and its vertical distance is y − k. By the Pythagorean theorem, the straight-line distance satisfies (x − h)² + (y − k)² = r², and the circle is exactly the set of points where this holds. Expanding the squares gives x² − 2hx + h² + y² − 2ky + k² = r², and moving everything to one side gives the general form. Going backwards uses completing the square: group the x terms and the y terms, add half of each coefficient squared, and the centre and radius reappear. If D²/4 + E²/4 − F is negative, no real circle exists; if it is zero, the "circle" is a single point.

Worked example

1. Inputs: centre (h, k) = (2, −3), radius r = 4.
2. Standard form: (x − 2)² + (y − (−3))² = 4², shown as (x − 2)² + (y + 3)² = 16.
3. D = −2 × 2 = −4; E = −2 × (−3) = 6.
4. F = 2² + (−3)² − 4² = 4 + 9 − 16 = −3.
5. General form: x² + y² − 4x + 6y − 3 = 0.

How to read the result

In the standard form, the signs inside the brackets are the opposite of the centre's coordinates: (y + 3) means k = −3. The number on the right is r², not r, so 16 means a radius of 4. In the general form, the coefficients of x and y are −2h and −2k, so dividing them by −2 recovers the centre. To test a point, substitute it: a result below 0 in the general form (or below r² in the standard form) means inside, 0 means on the circle, above means outside. The centre gives 4 + 9 − 8 − 18 − 3 = −16, so it is inside.

Common mistakes

• Reading the centre with the wrong signs from (x − 2)² + (y + 3)².
• Writing r instead of r² on the right side.
• Forgetting that x² and y² must have the same coefficient; if they differ, the curve is an ellipse, not a circle.

Frequently asked questions

How do I find the centre and radius of x² + y² − 4x + 6y − 3 = 0?

Complete the squares: (x − 2)² − 4 + (y + 3)² − 9 − 3 = 0, so (x − 2)² + (y + 3)² = 16, centre (2, −3), radius 4.

What if the x² and y² coefficients are 2 instead of 1?

Divide the whole equation by 2 first, then read D, E and F.

What is the equation of a circle centred at the origin?

x² + y² = r². D and E are zero and F = −r².

Sources: R. Descartes, La Géométrie (1637) · J. Stewart, L. Redlin and S. Watson, Precalculus: Mathematics for Calculus, Cengage, section on circles