How do I convert from standard form to general form?

Expand the squared terms: (x - h)² = x² - 2hx + h² and (y - k)² = y² - 2ky + k². Then combine like terms and move all terms to one side to get x² + y² + Dx + Ey + F = 0, where D = -2h, E = -2k, F = h² + k² - r².

How do I find the center and radius from general form?

Complete the square for x and y terms. The center is (h, k) where h = -D/2 and k = -E/2. The radius is r = √(h² + k² - F).

What if the circle is centered at the origin?

If the center is (0, 0), the standard form simplifies to x² + y² = r², and the general form is x² + y² - r² = 0 (where D = 0, E = 0, F = -r²).

Equation of a Circle Calculator

Calculate the equation of a circle in standard form (x-h)²+(y-k)²=r² and general form x²+y²+Dx+Ey+F=0. This calculator converts between center-radius form and general form.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

Equation of a Circle Calculator

Center X (h)

Center Y (k)

Radius (r)

How It Works

Standard form: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

General form: x² + y² + Dx + Ey + F = 0, where D = -2h, E = -2k, F = h² + k² - r².

To convert from standard to general form, expand and simplify the equation.

To convert from general to standard form, complete the square for both x and y terms.

The center can be found from general form: h = -D/2, k = -E/2, and r = √(h² + k² - F).

Step-by-step Examples

Example 1: Standard Form

Find the equation of a circle with center (3, -2) and radius 5.

Given: Center (h, k) = (3, -2), radius r = 5
Standard form: (x - h)² + (y - k)² = r²
Substitute: (x - 3)² + (y - (-2))² = 5²
Simplify: (x - 3)² + (y + 2)² = 25

Example 2: Converting to General Form

Convert (x - 3)² + (y + 2)² = 25 to general form.

Expand: (x - 3)² + (y + 2)² = 25
x² - 6x + 9 + y² + 4y + 4 = 25
Combine: x² + y² - 6x + 4y + 13 = 25
General form: x² + y² - 6x + 4y - 12 = 0
Where D = -6, E = 4, F = -12

Common Mistakes

Forgetting to square the radius in the standard form equation.
Making sign errors when converting between forms (especially with negative center coordinates).
Not completing the square correctly when converting from general to standard form.
Confusing the center coordinates - remember (h, k) means x = h, y = k.

Use Cases

Mathematics: Solving coordinate geometry problems involving circles.
Engineering: Describing circular paths and boundaries in coordinate systems.
Computer Graphics: Defining circular shapes and regions in 2D space.
Physics: Analyzing circular motion and trajectories.
Architecture: Specifying circular features in coordinate-based designs.

Frequently Asked Questions

Common questions about circle equations.

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