## Equation of a Circle Lesson

### Common Forms of a Circle's Equation

Let's take a look together through three common forms of a circle's equation (*standard form*, *general form*, and *polar coordinates*), so we can become more comfortable working with circles in algebra.

#### Standard Form for the Equation of a Circle

The standard form equation of a circle is given as:

**(x - a) ^{2} + (y - b)^{2} = r^{2}**

Where *a* is x-coordinate of the circle's center, *b* is the y-coordinate of the circle's center, and *r* is the radius of the circle.

Let's look at a circle defined by the equation (x - 1)^{2} + (y - 2)^{2} = 25. The center of this circle is located at the point (1, 2) since a = 1 and b = 2. The circle's radius is 5 since r^{2} = 25 and r = 5.

#### General Form for the Equation of a Circle

The general form equation of a circle is given as:

**x ^{2} + y^{2} + Ax + By + C = 0**

Where *A*, *B*, and *C* are constants.

The general form equation is not formatted to give us a quick glimpse of circle location and radius like the standard form equation is. However, general form can be converted to standard form by using the algebraic process known as completing the square.

#### Polar Coordinates Equation of a Circle

The standard form and general form of a circle's equation are expressed in the Cartesian coordinate system. **When dealing with circles, using the polar coordinate system is simpler than the Cartesian coordinate system.**

In polar coordinates, the equation of a circle centered over the origin is:

**r = radius of the circle**

If the circle is not centered over the origin but lies on the x-axis, the equation is:

**r = 2acosθ**

Where *a* is the x-coordinate of the circle's center.

If the circle is not centered over the origin but lies on the y-axis, the equation is:

**r = 2bsinθ**

Where *b* is the y-coordinate of the circle's center.