Geometry
Equation of a Circle Calculator
Radius and standard-form equation from center (h, k) and a point (x, y) on the circle.
Free to use — no sign-up or login.
Radius r
Equation (standard form)
Amortization schedule
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This free equation of a circle calculator reports radius r and the standard-form equation from a center (h, k) and one point (x, y) already on the circle. The defaults — h = 0, k = 0, x = 3, y = 4 — make a 3–4–5 right triangle, so r = 5 and the equation is (x − 0)² + (y − 0)² = 25, the same as x² + y² = 25. Here h and k are the center coordinates, r is the nonnegative radius, and the typed (x, y) is a sample point on the rim, not a second center.
No account is required. How it works beside the form keeps r = √((x − h)² + (y − k)²) and (x − h)² + (y − k)² = r² on separate lines. The Equation (standard form) field prints that squared identity with your numbers filled in. Expanding to x² + y² + Dx + Ey + F = 0 is a rewrite, not a second output on this page.
Standard form of a circle equation
In the Cartesian plane the standard form of a circle equation is (x − h)² + (y − k)² = r². The center is the point (h, k): h is the x-coordinate of the center and k is the y-coordinate. The radius r is the distance from that center to every point on the rim; it is a length, so this page never uses a negative r. In the written equation, x and y are variables that range over the circle. On this widget they are also the coordinates of one known point used to recover r. With the defaults, center (0, 0) and point (3, 4) give r = 5 and (x − 0)² + (y − 0)² = 25.
Radius from the center and a point
If a point (x, y) already lies on the circle, the radius is the Euclidean distance to the center: r² = (x − h)² + (y − k)², or r = √((x − h)² + (y − k)²). That is the same identity as the standard form, solved for r. Defaults: (3 − 0)² + (4 − 0)² = 9 + 16 = 25, so r = 5. Any other point at distance 5 from (0, 0) — for example (0, 5) or (−3, 4) — determines the same circle and the same equation, because the circle is the set of all points at that distance from the center, not a unique labeled point.
Standard form vs general form
Expand the standard form and collect terms: x² − 2hx + h² + y² − 2ky + k² = r². Moving everything to one side yields the general (expanded) form x² + y² + Dx + Ey + F = 0, sometimes written Ax² + Ay² + Dx + Ey + F = 0 with equal quadratic coefficients. The matches are D = −2h, E = −2k, and F = h² + k² − r². For the default circle that is x² + y² − 25 = 0. This calculator prints the compact standard form (x − 0)² + (y − 0)² = 25, not the expanded general polynomial. Completing the square takes you back from general form to (h, k) and r.
How to use this calculator
Type the center in Center h (x) and Center k (y), default 0 and 0, then a point already on the circle in Point on circle x and Point on circle y, default 3 and 4. Radius r updates as the distance; Equation (standard form) substitutes those numbers into (x − h)² + (y − k)² = r². You do not enter r on this page — the point supplies it. Disk area and circumference from a known r are the Area of a Circle Calculator and the How to Calculate Circle Area and Circumference tutorial, not this equation widget.
Unit circle, r = 0, and why a circle is not a function
If the typed point equals the center, the distance is zero: r = 0 and the “circle” is a single point, (x − h)² + (y − k)² = 0. That degenerate case is allowed as arithmetic, not as a positive-radius circle. Radius is never entered as a negative number here; hypot(x − h, y − k) is ≥ 0. The unit circle is the special case h = 0, k = 0, r = 1, so x² + y² = 1 — pick a point such as (1, 0) or (0, 1). A circle is not the graph of a function y = f(x): a vertical line through an interior x usually meets the rim twice (upper and lower semicircles), and a horizontal line does the same in x. Solving the standard form for y yields two branches, y = k ± √(r² − (x − h)²), not a single output.
Related circle formulas
Area πr² and circumference 2πr from a known radius are the Area of a Circle Calculator. A half-disk is the Semicircle Area Calculator. A tangent segment from an external point to a circle about the origin is the Tangent of a Circle Calculator. Sine and cosine as a point on x² + y² = 1 are the Unit Circle Calculator. The same r in three dimensions is the Sphere Calculator.
Worked example: center (0, 0) and point (3, 4)
Center h = 0, k = 0, and a point on the circle x = 3, y = 4. Then r² = (3 − 0)² + (4 − 0)² = 9 + 16 = 25, so r = 5. Substitute into standard form: (x − 0)² + (y − 0)² = 25, which is x² + y² = 25. That is the equation this widget prints for the defaults. The same r and equation appear for any other rim point at distance 5 from the origin; the point only has to lie on the circle.
\[ r=\sqrt{3^{2}+4^{2}}=5,\quad (x-0)^{2}+(y-0)^{2}=25 \]
Frequently asked questions
What is the standard equation of a circle?
Standard form is (x − h)² + (y − k)² = r², where (h, k) is the center, r is the radius, and (x, y) runs over points on the circle. With the defaults h = 0, k = 0, r = 5 that is x² + y² = 25.
How do I find the radius from the center and a point?
Take the distance: r = √((x − h)² + (y − k)²). For center (0, 0) and point (3, 4) that is √(9 + 16) = 5. This calculator does that from the four coordinate fields.
What is the difference between standard form and general form?
Standard form keeps the completed squares (x − h)² + (y − k)² = r². General form expands that to x² + y² + Dx + Ey + F = 0 (equal coefficients on x² and y²). This page outputs standard form; for the defaults that is (x − 0)² + (y − 0)² = 25, not x² + y² − 25 = 0.
What happens if the point is the same as the center?
Then r = 0: the equation is (x − h)² + (y − k)² = 0, a single point rather than a circle with positive radius. A negative radius is not used; distance is never negative.
What is the equation of the unit circle?
The unit circle is center (0, 0) and r = 1, so x² + y² = 1. On this page set h = 0, k = 0 and choose a point at distance 1, such as (1, 0). Cosine and sine as coordinates on that circle are the Unit Circle Calculator.
Is a circle a function of x?
No. Except in degenerate r = 0 cases, a vertical line through an x between h − r and h + r meets the circle at two y-values. The graph fails the vertical line test, so y is not a single-valued function of x; the two semicircles y = k ± √(r² − (x − h)²) are functions, the full circle is not.
References
Assumptions and limitations
These circle-equation results are unofficial Euclidean-plane identities. The radius is a length (≥ 0). Coordinate units are yours to name; they must match for x, y, h, and k.