This circle is centered at the origin, and the length of its radius is 6. What is the circle's equation?

1. ### Break Down the Problem

To find the equation of a circle centered at the origin with a given radius, we need to use the standard equation of a circle. The center of the circle and how the radius relates to the coordinates will help us derive the equation.

2. ### Relevant Concepts

The standard form of the equation of a circle centered at the origin $(0, 0)$ with radius $r$ is given by: $x^2 + y^2 = r^2$

3. ### Analysis and Detail

Given that the radius $r = 6$, we can substitute this value into the standard circle equation.

1. First, calculate $r^2$: $r^2 = 6^2 = 36$

2. Now, substitute $r^2$ into the circle's equation: $x^2 + y^2 = 36$

4. ### Verify and Summarize

We established the basic formula for a circle with given radius and confirmed our calculations for $r^2$ were accurate.

Final Answer

The equation of the circle is: $x^2 + y^2 = 36$