Equations of Circles in the Coordinate Plane Cheat Sheet

Key Facts

• The center-radius form of a circle is $\left(x - h\right)^2 + \left(y - k\right)^2 = r^2$, where the center is $\left(h,k\right)$ and the radius is $r$.
• A circle centered at the origin has equation $x^2 + y^2 = r^2$.
• If the equation is $\left(x + 3\right)^2 + \left(y - 5\right)^2 = 16$, the center is $\left(-3,5\right)$ and the radius is $4$.
• The distance formula $d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}$ can be used to find a circle’s radius from its center and a point on the circle.
• The diameter is twice the radius, so $d = 2r$, and the radius is half the diameter, so $r = \frac{d}{2}$.
• To complete the square for $x^2 + bx$, add $\left(\frac{b}{2}\right)^2$ to make $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$.
• A general circle equation has the form $x^2 + y^2 + Dx + Ey + F = 0$, with equal coefficients on $x^2$ and $y^2$ and no $xy$ term.
• After rewriting in center-radius form, the value on the right must be positive because $r^2 > 0$ for a real circle.

Vocabulary

Circle

A circle is the set of all points in a plane that are the same distance from one fixed point.

Center

The center is the fixed point $\left(h,k\right)$ that is the same distance from every point on the circle.

Radius

The radius is the distance $r$ from the center of a circle to any point on the circle.

Diameter

The diameter is a segment through the center with endpoints on the circle, and its length is $2r$.

Center-Radius Form

Center-radius form is $\left(x - h\right)^2 + \left(y - k\right)^2 = r^2$, which shows the center and radius directly.

Completing the Square

Completing the square is an algebraic method that rewrites a quadratic expression as a perfect square trinomial.