Circle Equation Calculator

Find standard circle equation from center and radius. This educational calculator shows the formula, result, and step-by-step interpretation.

High SchoolGeometry

Calculator

What this calculator teaches

Circle equations are a core part of coordinate geometry and conic sections.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

Reading Center and Radius from Standard Form

A circle is the set of all points whose distance from a fixed center (h,k) equals a constant radius r. Starting from the distance formula and squaring both sides gives (x-h)2+(y-k)2=r2. The squared terms are what make every direction from the center behave symmetrically.

The signs inside the parentheses are opposite the center coordinates. A factor (x-2)2 means h=2, while (y+3)2 is the same as (y-(-3))2, so k=-3. The right side is the radius squared, not the radius itself.

PartMeaningHow to read it
(x-h)2Horizontal displacement squaredCenter x-coordinate is h
(y-k)2Vertical displacement squaredCenter y-coordinate is k
r2Squared radiusRadius is the nonnegative square root

A point (x,y) lies on the circle exactly when substitution makes the left side equal r2. If a circle is given in expanded general form, completing the square in x and y can recover the standard form, center, and radius.

Another quick interpretation comes from symmetry. Moving r units horizontally or vertically from the center produces the four points (h±r,k) and (h,k±r). Substituting any of those points makes one squared displacement r2 and the other zero, confirming the standard equation immediately.

Formula & Symbols

ConceptFormula or rule
Circle equation(x-h)² + (y-k)² = r²

Worked example

Example: Center (2,-3) and radius 5 gives (x-2)² + (y+3)² = 25.
Example 2: Center (0,0) with radius 4 gives x2+y2=16.
Example 3: Center (-2,3) with radius 2 gives (x+2)2+(y-3)2=4. The plus sign in x corresponds to a negative center coordinate.
Example 4: Center (5,-1) with radius 0 gives (x-5)2+(y+1)2=0, a degenerate circle consisting only of the center point.
Example 5: The general equation x2+y2-6x+4y-12=0 completes to (x-3)2+(y+2)2=25, so its center is (3,-2) and radius is 5.

Common mistakes

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Using the wrong input format

Keep lists comma separated, matrices as rows separated by semicolons, and modular inputs as integers.

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Ignoring assumptions

Some methods require positive probabilities, valid moduli, independent trials, or small educational input sizes.

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Reading the center signs directly from the parentheses

In (x-h)2+(y-k)2, the center is (h,k). Thus (x+4)2 means h=-4, not +4.

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Putting r instead of r squared on the right

A radius of 5 produces a right side of 25. The equation comes from the squared distance formula, so the constant is r2.

FAQ

Can I use decimals?⌄
Most numerical calculators allow decimals, but modular arithmetic and coding-theory tools usually require integers or binary strings.
Is this for homework checking?⌄
Yes. The page is designed to show both the answer and the reasoning pattern.
Why does the result sometimes say approximate?⌄
Some probability, floating-point, and numerical methods naturally produce approximations.
How do I identify the center from a circle equation?⌄
Match the equation to (x-h)2+(y-k)2=r2. The numbers h and k are the center coordinates, remembering that a plus sign inside a parenthesis corresponds to a negative coordinate.
How can I test whether a point lies on the circle?⌄
Substitute the point's x and y coordinates into the left side. If the sum equals r2, the point is on the circle; a smaller sum is inside and a larger sum is outside.
What is the difference between standard and general circle form?⌄
Standard form displays center and radius directly. General form expands the squares, typically producing x2+y2+Dx+Ey+F=0. Completing the square converts the general form back to standard form.
Can a circle have a negative radius?⌄
No in ordinary geometry; radius is a distance and is nonnegative. Squaring would hide the sign algebraically, so radius inputs are conventionally restricted to r≥0.

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Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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