About this calculator
Circle Equation Calculator helps students, teachers, and practical users work with coordinate geometry, algebra, graphing, and sequence problems. Use it to convert between centre-radius form and general form for a circle equation. The calculator is built for learning as well as quick checking, so it shows the key formula, the main result, and supporting values that help explain the answer.
Circle Equation Calculator method
The calculator expands standard form or completes the square in general form to find the centre and radius.
- standard form: (x - h)^2 + (y - k)^2 = r^2
- general form: x^2 + y^2 + Dx + Ey + F = 0
- centre = (-D / 2, -E / 2)
How to use the Circle Equation Calculator
- Enter the points, equation, coefficients, or sequence values requested by the calculator.
- Check signs carefully, especially negative coordinates, negative gradients, and standard-form coefficients.
- Review the highlighted result first, then use the supporting values to understand how it was found.
- Compare the equation forms or formula breakdown if your homework asks for working.
- Change one value at a time when comparing different lines, curves, or sequences.
- Round only at the final answer unless your worksheet or exam question gives a rounding rule.
Worked examples
Standard to general
Input: Centre (2, -3), radius 5.
Calculation: (x - 2)^2 + (y + 3)^2 = 25 expands to x^2 + y^2 - 4x + 6y - 12 = 0.
Result: General form: x^2 + y^2 - 4x + 6y - 12 = 0.
General to centre
Input: x^2 + y^2 - 4x + 6y - 12 = 0.
Calculation: Centre is (2, -3). Radius squared is 25.
Result: Standard form: (x - 2)^2 + (y + 3)^2 = 25.
What this calculator is useful for
Circle equations are common in coordinate geometry, graph sketching, and completing-the-square practice.
For UK maths work, the word gradient is often used where US resources say slope. The calculation is the same: rise divided by run, or change in y divided by change in x.
Common notation
| Notation | Meaning |
|---|---|
| m | Slope or gradient |
| b | Y-intercept in y = mx + b |
| Ax + By = C | Standard form of a line |
| x1, y1 | Coordinates of the first point |
| n | Term number in a sequence |
When to show extra working
If the answer is for homework, an exam-style question, or a worksheet, include the formula substitution as well as the final value. For example, write the rise and run before simplifying the gradient, or show the completed square when converting a circle equation.
Common mistakes and edge cases
- A vertical line has undefined slope because the run is zero.
- A horizontal line has slope 0 and may not have an x-intercept unless it is y = 0.
- Negative signs in coordinates and equations are the most common source of wrong answers.
- Equivalent equation forms can look different but describe the same line or curve.
- Sequence formulas assume the pattern is truly arithmetic or geometric.
Limitations
This calculator is for general educational information only. It follows standard school-level algebra and coordinate geometry methods, but your course, teacher, or software may prefer a specific form, rounding convention, or notation.
- Check whether your answer must be exact, decimal, fractional, or rounded.
- Equation parsing supports common typed forms and may not understand every possible algebra layout.
- For high-stakes engineering, design, or technical work, verify calculations independently.
Frequently asked questions
Is gradient the same as slope?
Yes. In UK maths, gradient is the common term. In many US resources, the same calculation is called slope.
What is y = mx + b?
It is slope-intercept form. m is the gradient and b is the y-intercept.
What happens if the line is vertical?
The gradient is undefined because x does not change, so the run is zero.
Why do equivalent equations look different?
A line can be written in slope-intercept, point-slope, or standard form. The forms are algebraically equivalent.
Should I round the result?
Use exact values where possible and round at the end according to the question instructions.
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