Put Circle in Standard Form Calculator

What is Standard Form of a Circle?

The standard form of a circle equation is a way to represent a circle's position and size in a coordinate plane. It's written as:

Where:

• (h, k) are the coordinates of the circle's center
• r is the radius of the circle

This form makes it easy to identify the circle's center and radius directly from the equation.

How to Convert a Circle to Standard Form

To convert any circle equation to standard form, follow these steps:

1. Start with the general form of a circle equation: x² + y² + Dx + Ey + F = 0
2. Rearrange the terms to group x and y terms together
3. Complete the square for both x and y terms
4. Factor out the coefficients of the squared terms
5. Write the equation in the standard form (x - h)² + (y - k)² = r²

This process involves algebraic manipulation to identify the center and radius.

The Formula

The general form of a circle equation is:

To convert to standard form:

1. Rearrange: x² + Dx + y² + Ey = -F
2. Complete the square for x: (x + D/2)² - (D/2)²
3. Complete the square for y: (y + E/2)² - (E/2)²
4. Combine terms: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F
5. Final standard form: (x - h)² + (y - k)² = r²

Where h = -D/2, k = -E/2, and r² = (D/2)² + (E/2)² - F

Worked Example

Let's convert the equation x² + y² - 6x + 8y + 9 = 0 to standard form.

1. Rearrange: x² - 6x + y² + 8y = -9
2. Complete the square for x: (x² - 6x + 9) - 9 + y² + 8y = -9 → (x - 3)² - 9 + y² + 8y = -9
3. Complete the square for y: (x - 3)² - 9 + (y² + 8y + 16) - 16 = -9 → (x - 3)² + (y + 4)² - 25 = -9
4. Combine constants: (x - 3)² + (y + 4)² = 16

The standard form is (x - 3)² + (y + 4)² = 16, with center at (3, -4) and radius 4.

Interpreting the Result

Once you have the circle in standard form, you can easily identify:

• The center (h, k) from the terms (x - h) and (y - k)
• The radius r from the term r²

This information helps with graphing, analyzing the circle's position, and solving related problems.