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 = r2
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 r2 = 25 and r = 5.
General Form for the Equation of a Circle
The general form equation of a circle is given as:
x2 + y2 + 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.