Math

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Standard to General Form Circle Calculator.

Expand a circle's standard equation into general polynomial form.

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Use Cases

Algebra homework and practice

Quickly verify your manual expansion of standard form to general form, ensuring your coefficients and constant term are correct.

Example: For center (3, -2) and radius 5, the general form is x^2 + y^2 - 6x + 4y - 12 = 0.

Graphing and equation conversion

When you need to switch between forms for graphing or solving problems, this tool gives you the general form instantly.

Example: Convert (x+1)^2 + (y-4)^2 = 9 to x^2 + y^2 + 2x - 8y + 8 = 0.

Frequently Asked Questions

What does this calculator do?
It takes the center (h, k) and radius (r) of a circle in standard form and expands the equation into general form: x^2 + y^2 + Dx + Ey + F = 0. You input h, k, and r, and it computes the coefficients.
How is the general form derived?
Starting from (x - h)^2 + (y - k)^2 = r^2, expand the squares, move all terms to one side, and combine constants. The result is x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0.
Can I use this for a circle not centered at the origin?
Yes. Enter any values for h and k (including negative numbers). The calculator handles any center, not just (0,0).

Tips & Common Mistakes

Tips

  • Ensure the radius is positive; if you enter a negative radius, the calculator may still work but the equation is invalid for a real circle.
  • Double-check the signs of h and k. For example, if the center is (-2, 3), enter h = -2 and k = 3.
  • Use the general form to easily identify the coefficients D, E, and F, which are useful for further algebraic manipulations.
  • Remember that the general form is not unique if you multiply the entire equation by a constant; this calculator gives the standard form with leading coefficient 1.

Common Mistakes to Avoid

  • Forgetting to square the radius when moving it to the left side. The constant term is h^2 + k^2 - r^2, not h^2 + k^2 - r.
  • Misplacing signs when expanding (x - h)^2. For negative h, the term becomes +2|h|x, not -2hx.
  • Assuming the general form always has integer coefficients; if h, k, or r are decimals, the coefficients will be decimals.

Last updated: August 13, 2026