Standard Form of a Circle Equation
The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form makes it easy to identify the center and radius of the circle directly from the equation.
Circle Equation Forms
Standard Form
The most useful form, directly showing center and radius.
(x - h)² + (y - k)² = r²
General (Expanded) Form
The expanded polynomial form: x² + y² + Dx + Ey + F = 0.
x² + y² + Dx + Ey + F = 0
Unit Circle
A special circle centered at the origin with radius 1.
x² + y² = 1
How to Convert Between Forms
Standard to General Form
- Expand (x - h)² to get x² - 2hx + h².
- Expand (y - k)² to get y² - 2ky + k².
- Add both expansions and set equal to r².
- Move r² to the left side and combine constants: D = -2h, E = -2k, F = h² + k² - r².
Key Properties
- The center (h, k) is the point equidistant from all points on the circle.
- The radius r must be a positive number.
- Area = pi x r² and circumference = 2 x pi x r.
- Any point (x, y) on the circle satisfies the equation exactly.