The correct equation that relates the circumference and area of a circle is not explicitly listed among the options you've provided. However, we can recall the well-known formulas for the circumference \( C \) and area \( A \) of a circle, which are:
- Circumference: \( C = 2\pi r \)
- Area: \( A = \pi r^2 \)
From these formulas, we can derive relationships between the area and circumference, but none of the given options accurately represent these relationships.
For clarity:
- \( A = \frac{C \cdot r}{2\pi} \) implies a derived relationship, but it's not one of the provided options.
- \( C = 2\sqrt{A\pi} \) is another derived relation from these formulas.
None of the options correctly express a valid formula relating the area and circumference directly. If you are looking for an equation that shows the relationship between the area \( A \) and circumference \( C \) that incorporates the radius \( r \), it's important to note that the equations provided do not form valid relationships, as they are not consistent with the formulas above.