The equation of a circle with center (h, k) and radius r is (x − h)2 + (y − k)2 = r2. It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x2 + y2 = r2.
In Spanish
ecuación de una circunferencia
Also called
standard form of a circle, center-radius form, circle equation
The equation is the Pythagorean theorem for the triangle from the center to any point on the circle: the legs are x − h and y − k, and the hypotenuse is always the radius r.
Formal definition and notation
A circle is the set of all points in a plane at a fixed distance r > 0, the radius, from a fixed point (h, k), the center. By the distance formula, which is the Pythagorean theorem on the coordinate plane, a point (x, y) lies on the circle exactly when √(x − h)2 + (y − k)2 = r. Squaring both sides gives the standard form(x − h)2 + (y − k)2 = r2. Multiplying out gives the general form x2 + y2 + Dx + Ey + F = 0, which you turn back into standard form by completing the square.
Circumference is a length, the distance around the circle, 2π r. The equation of a circle is a rule that every point on the circle satisfies. For x2 + y2 = 25, the circumference is 10π.
Equation of a Circlevs.Unit circle
The unit circle is one special circle: center (0, 0) and radius 1, so its equation is x2 + y2 = 1. It is the circle used in trigonometry.
Equation of a Circlevs.Parabola
A parabola such as y = x2 squares only one variable. A circle squares both x and y, with equal coefficients.
Quick check: 3 questions
Question 1 of 3
What is the center of (x − 7)2 + (y + 3)2 = 64?
Show answer and explanation
Answer: B.x − 7 gives h = 7, and y + 3 = y − (−3) gives k = −3. (−7, 3) flips both signs, (7, 3) flips one, and (−3, 7) swaps the coordinates.
Question 2 of 3
What is the radius of x2 + y2 = 50?
Show answer and explanation
Answer: C.r2 = 50, so r = √50 = √25 × 2 = 5√2 ≈ 7.07. 50 is r2, 25 halves it, and 10 comes from 505.
Question 3 of 3
Which point lies on the circle (x − 1)2 + (y − 2)2 = 25?
Show answer and explanation
Answer: D. For (4, 6): 32 + 42 = 9 + 16 = 25. For (6, 7) the sum is 50 (outside), for (3, 4) it is 8 (inside), and for (5, 2) it is 16 (inside).
Equation of a Circle: frequently asked questions
How do you find the equation of a circle from its center and one point?
Use the distance from the center to the point as the radius. For center (2, −1) and point (5, 3): r2 = (5 − 2)2 + (3 − (−1))2 = 9 + 16 = 25. You do not even need r itself, only r2, so the equation is (x − 2)2 + (y + 1)2 = 25.
How do you find the center and radius from the general form?
Group the x terms and the y terms, move the constant to the right, and complete the square for each group, adding the same numbers to both sides. Then read off the center and take the square root of the right side. If the right side comes out zero or negative, the equation is a single point or has no graph at all.
Is a circle a function?
No. A function gives each input one output, but most x values on a circle have two y values, one on the top half and one on the bottom. For x2 + y2 = 25, the input x = 3 gives y = 4 and y = −4. The top half alone, y = √25 − x2, is a function.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSG-GPE.A.1
Cite this entry
Effortless Math Team. “Equation of a Circle.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/equation-of-a-circle/
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