Equation of a Circle Calculator
Find the standard form, general form, center, radius, and area of a circle from various starting conditions.
Choose your given information
An equation of a circle calculator finds a circle’s equation from its center and radius or from points on the circle. A circle with center (h, k) and radius r uses the equation (x - h)² + (y - k)² = r².
A circle is perfectly defined by its center point (h, k) and its radius r. Select the information you have from the dropdown below to calculate the complete equation of the circle in both standard and general forms.
Standard Form
General Form
Parametric Form
y = k + r sin(t)
Circle Properties
Center
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Radius (r)
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Diameter
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Area
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Circumf.
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X-Intercepts
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Y-Intercepts
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Calculation Steps
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How to use the circle equation calculator
- Select your given info: Use the dropdown to choose what parameters you know (e.g., Center and Radius).
- Enter the values: Type the coordinates, length, or coefficients into the corresponding fields. Decimals and negative numbers are supported.
- Calculate: Press the "Calculate Circle" button to generate the standard and general equations.
- Review properties: The calculator automatically determines the circle's center, radius, diameter, exact area, and circumference.
Circle Equations & Formulas
1. Standard Form
The standard form equation of a circle is derived directly from the Pythagorean theorem (or distance formula). It clearly displays the center coordinates (h, k) and the radius r.
(x - h)² + (y - k)² = r²
2. General Form
The general form is the expanded version of the standard form. You can convert from standard to general form by expanding the binomials and moving all terms to one side.
x² + y² + Dx + Ey + F = 0
To find the center from the general form, use: h = -D/2 and k = -E/2.
The radius is found using: r = √(h² + k² - F).
How to Convert General Form to Standard Form
Converting a circle's equation from general form (x² + y² + Dx + Ey + F = 0) to standard form requires a mathematical technique called completing the square. Here is a step-by-step example using the equation: x² + y² + 4x - 6y - 3 = 0.
Step 1: Group x and y terms together
Move the constant (F) to the right side of the equation.
(x² + 4x) + (y² - 6y) = 3
Step 2: Complete the square for both variables
Take half of the x coefficient, square it, and add it to both sides. Do the same for y.
- For x: Half of 4 is 2. 2² = 4.
- For y: Half of -6 is -3. (-3)² = 9.
(x² + 4x + 4) + (y² - 6y + 9) = 3 + 4 + 9
Step 3: Factor the perfect square trinomials
Rewrite the grouped terms as squared binomials and simplify the right side to find r².
(x + 2)² + (y - 3)² = 16
Result: We can now clearly see the center is (-2, 3) and the radius is 4 (since √16 = 4).
Finding the X and Y Intercepts of a Circle
The intercepts are the exact coordinates where the circle crosses the x-axis and y-axis. Depending on its position and radius, a circle can have zero, one, or two intercepts on each axis.
X-Axis How to find X-Intercepts
The x-intercepts occur where the circle crosses the horizontal axis. This means the height is zero (y = 0).
- Substitute 0 for y in your standard equation.
- Solve the resulting quadratic equation for x:
(x - h)² + (0 - k)² = r² - Isolate x by taking the square root of both sides.
Y-Axis How to find Y-Intercepts
The y-intercepts occur where the circle crosses the vertical axis. This means the horizontal distance is zero (x = 0).
- Substitute 0 for x in your standard equation.
- Solve the resulting quadratic equation for y:
(0 - h)² + (y - k)² = r² - Isolate y by taking the square root of both sides.
Note on missing intercepts: If calculating the square root during steps 2 or 3 results in a negative number under the radical (e.g., √-4), the circle does not intersect that particular axis.
Beyond the Basics: Advanced Circle Anatomy
Secant Line
An infinite line that intersects a circle at exactly two distinct points, passing straight through its interior.
Tangent Line
A straight line that touches the circle at exactly one single point on its outer edge, never crossing into the interior.
Arc
A continuous curve representing a portion of the circumference. A "major arc" is larger than a semicircle; a "minor arc" is smaller.
Sector
A pie-shaped geometric region of a circle enclosed by two radii and the arc connecting them on the perimeter.
Chord
A straight line segment whose endpoints lie entirely on the circle. (A diameter is simply the longest possible chord).
Concentric
Two or more separate circles that share the exact same geometric center point (h, k) but have different radius lengths.
Frequently Asked Questions
How do I find the equation of a circle with two points?
If the two points on the Cartesian plane represent the outer endpoints of a diameter, you can easily find the center by calculating the midpoint between them. The radius is simply half the distance between those two locations. Once you determine the center and radius, you can enter them into our calculator or manually write the standard form equation of the circle.
Why is the radius squared in the standard equation?
The standard formula is directly based on the distance formula (derived from the Pythagorean theorem in geometry). The distance from any given point (x, y) on the circle's edge back to the center (h, k) is exactly the radius 'r'. In math, squaring both sides of this distance equation neatly removes the square root, giving us the highly useful format (x - h)² + (y - k)² = r².
What if the radius squared (r²) is negative?
If you are evaluating a general form equation and calculate that r² is less than 0, it means the expression represents an imaginary circle. There is no real-world solution and no points will satisfy this equation, meaning you cannot graph it on a standard 2D plane. If r² equals exactly 0, the equation mathematically represents a single point (the center) rather than a full shape.
How do I find the center and radius from the general form?
To find the hidden center coordinate (h, k) and radius r from the general form x² + y² + Dx + Ey + F = 0, you can use these shortcut mathematical formulas: h = -D/2, k = -E/2, and r = √(h² + k² - F). Alternatively, you can use the completing the square method to rewrite it into standard form. Or, simply plug your D, E, and F values into the calculator above for an instant solution.
What is the equation of a circle centered at the origin?
If a circle is centered exactly at the origin (0, 0) on your graph, the standard form equation simplifies significantly. Because the x-coordinate (h) is 0 and the y-coordinate (k) is 0, the center terms drop out entirely, and the final equation becomes simply x² + y² = r².
How do you determine if a point is inside, outside, or on the circle?
Take the specific x-coordinate and y-coordinate of your test point and plug them into the left side of the standard form equation: (x - h)² + (y - k)². If the calculated result is exactly equal to r², the point lies perfectly on the circle's edge (it is a valid solution). If the result is less than r², the point is located inside the shape. If it is greater than r², it lies completely outside.
Can a circle have a radius of zero?
In geometry, a shape with a radius of exactly zero is classified as a degenerate circle. When you try to graph this, it appears as only a single individual coordinate—its center—having no area or measurable circumference. Its mathematical equation would simply end with "= 0" instead of a positive r².
Why do we use completing the square for circle equations?
In math, completing the square is a vital technique that allows us to neatly group the scattered x and y variables from the general form into perfect square binomials. This specific algebraic formula manipulation is required to convert the expanded format back into the standard equation, which clearly reveals the shape's true center and radius for accurate graphing.
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Disclaimer: This equation of a circle calculator is for general educational and informational use. It relies on standard float precision arithmetic, meaning highly complex decimal inputs may result in slightly rounded outputs. Always double-check your mathematical results.
Last updated: May 20, 2026