Equation for Circle- Standard Form, General Form, and Examples

What Is the Equation for a Circle?

A circle is the set of all points in a plane that are equidistant from a fixed point. That fixed point is the center, and the distance from the center to any point on the circle is the radius.

The equation of a circle is just a way of expressing that relationship algebraically. You need two things: where the center sits and how big the radius is.

There are two main forms you'll encounter — standard form and general form. Each has its own use depending on what you're trying to do.

Standard Form of a Circle Equation

The standard form looks like this:

(x - h)² + (y - k)² = r²

Where:

• (h, k) is the center of the circle
• r is the radius

The equation says: "Any point (x, y) that is exactly r units away from (h, k) is on this circle."

Reading Standard Form

Take (x - 3)² + (y + 2)² = 25

The center is at (3, -2). Notice the sign flip — it's (x - 3) so h = 3, and (y + 2) means k = -2.

The radius is √25 = 5. You square the radius in the equation, so you take the square root to get the actual radius.

Why This Form Exists

Standard form makes it obvious where the circle is centered and how large it is. You can sketch it immediately without calculating anything. That's the main advantage.

General Form of a Circle Equation

The general form expands everything out:

x² + y² + Dx + Ey + F = 0

There are no parentheses and no squared terms grouped together. Everything is written as a polynomial equal to zero.

Example in General Form

x² + y² - 6x + 4y - 12 = 0

You can't tell the center or radius just by looking. You have to complete the square to extract that information.

When General Form Shows Up

General form appears when you multiply out standard form or when the equation comes from raw data. It's less intuitive but sometimes easier to work with in certain algebraic manipulations.

How to Find Center and Radius from Any Form

From Standard Form

This is straightforward. Just read off h, k, and r.

Example: (x + 1)² + (y - 5)² = 16

• Center: (-1, 5)
• Radius: 4

From General Form — Complete the Square

This is where most people get stuck. Here's the process:

Given: x² + y² + Dx + Ey + F = 0

Step 1: Group x terms and y terms

x² + Dx + y² + Ey + F = 0

Step 2: Complete the square for each group

Take x² + Dx. Add (D/2)² to both sides. Take y² + Ey. Add (E/2)² to both sides.

Step 3: Rewrite as standard form and read off center and radius

Worked Example

Convert x² + y² - 6x + 4y - 12 = 0 to standard form.

Step 1: Group terms

(x² - 6x) + (y² + 4y) = 12

Step 2: Complete the squares

x² - 6x → add 9 (because (-6/2)² = 9)

y² + 4y → add 4 (because (4/2)² = 4)

(x² - 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4

Step 3: Factor

(x - 3)² + (y + 2)² = 25

Center: (3, -2). Radius: 5.

Converting Between Forms

Going from standard to general is just expansion:

(x - h)² + (y - k)² = r²

Expand: x² - 2hx + h² + y² - 2ky + k² = r²

Rearrange: x² + y² - 2hx - 2ky + (h² + k² - r²) = 0

Going from general to standard requires completing the square, as shown above. That's the only method that works reliably.

Comparison: Standard Form vs General Form

Equation of a Circle with Center at Origin

When the center is at (0, 0), the equation simplifies to:

x² + y² = r²

That's it. No h or k to worry about.

Examples:

• x² + y² = 9 → circle centered at origin with radius 3
• x² + y² = 2 → circle centered at origin with radius √2

Common Mistakes to Avoid

• Ignoring sign flips: (x - 3) means center at x = 3, but (x + 3) means center at x = -3. The sign inside the parentheses is opposite the actual coordinate.
• Forgetting to square root: The radius is squared in the equation. Take the square root to get the actual radius.
• Messy completing the square: Whatever you add to one side, add to the other. People drop terms constantly here.

Getting Started: How to Write a Circle Equation

Given center (h, k) and radius r:

1. Write (x - h)² + (y - k)² = r²
2. Substitute your values
3. Simplify if needed

Example: Center (2, 5), radius 3

(x - 2)² + (y - 5)² = 9

That's the answer. No further work needed.

Given three points on the circle:

1. Set up three equations using x² + y² + Dx + Ey + F = 0
2. Solve the system for D, E, and F
3. Complete the square to find center and radius

This is more involved and usually shows up in problems where you need to find the unique circle passing through three points.

Quick Reference

• Standard form: (x - h)² + (y - k)² = r² — center and radius are obvious
• General form: x² + y² + Dx + Ey + F = 0 — complete the square to extract center and radius
• Origin-centered: x² + y² = r² — special case with h = 0, k = 0
• Completing the square: The tool for converting general form to standard form

Know these two forms and how to move between them. That's the entire circle equation topic in one sentence.