Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry equation of a circle? Then this Geometry lesson on Writing and Graphing Circle Equations explains each idea step by step, with worked examples.
Quick answer: The equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². It comes from the Pythagorean Theorem applied to the horizontal and vertical distances from the center. Completing the square converts any general circle equation into this form.
Standards: G-GPE.A.1 · Level: Grade 10 Geometry
Where does the equation of a circle come from?
Think of a circle as every point that stays exactly r units away from a fixed spot (h, k). Pick a point (x, y) on it. Walking from the center to that point takes a horizontal step of |x − h| and a vertical step of |y − k|, and the radius closes a right triangle.
(x − h)2 + (y − k)2 = r2
The center is (h, k) and the radius is r. A circle centered at the origin is x2 + y2 = r2.
How do you read the center and radius?
Signs flip: (x + 3) hides h = −3, while (y − 5) hides k = 5. The constant on the right side equals r2, so a square root is still needed.
How do you rewrite a general equation by completing the square?
- Collect the x-terms together, the y-terms together, and send the constant across the equals sign.
- Square half of each linear coefficient and add both results to each side.
- Turn each trinomial into a squared binomial.
- Read off the center and radius, confirming the right side is positive.
Rewrite x² + y² − 10x + 4y − 7 = 0.
(x² − 10x) + (y² + 4y) = 7. Add 25 and 4: (x − 5)² + (y + 2)² = 7 + 25 + 4 = 36.
Center (5, −2), radius 6.
How do you tell whether a point is inside, on or outside?
| Compute (x − h)² + (y − k)² and compare with r² | Position |
|---|---|
| less than r² | inside |
| equal to r² | on the circle |
| greater than r² | outside |
Using the wrong sign for the center, or forgetting to take the square root of the right side to get the radius.
- Standard form: (x − h)² + (y − k)² = r², center (h, k), radius r.
- A circle centered at the origin is x² + y² = r².
- Complete the square in x and in y to read the center and radius from a general equation.
- A point is inside, on or outside the circle when its squared distance is less than, equal to or greater than r².
Writing and Graphing Circle Equations: frequently asked questions
Why does the circle equation use squares?
Every point on the circle is a distance r from the center, and distance comes from the Pythagorean Theorem. Squaring the horizontal and vertical differences avoids square roots in the equation.
How do I find the center from (x + 4)² + (y − 7)² = 9?
Reverse each sign. The center is (−4, 7), because x + 4 equals x − (−4). The radius is √9 = 3.
What if the squared terms have a coefficient larger than 1?
Divide the whole equation by that coefficient first so that x² and y² each have coefficient 1, then complete the square.
Can a circle equation have no graph?
Yes. After completing the square, a negative number on the right means no real points satisfy the equation, and zero means the graph is a single point.
How do I write the equation from a diameter?
Find the midpoint of the diameter for the center, then use the distance from the center to an endpoint for the radius.
Mis à jour le